Why certain diophantine equations are interesting (and others are not) ? It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" mathematical generalizations thereof, others come from geometry in a variety of ways. 
For diophantine equations there seem not to be such a direct link to other areas.
I would like to roughly understand why the attention of number theorists concentrates on some kinds of diophantine equations and not on others. 
Why an equation such as 
$x^2-ny^2=1$
or
$x^3+y^3=z^3$
is (or have been) considered worth studying, and not, say, any other random variant such as (if that specific example is not enough nontrivial for you or if it actually happens to have been studied, feel free to substitute it with your favourite "random" diophantine equation):
$x^3+y^5=z^2$  ?  So:

Are there any reasons why certain diophantine equations are worth attention besides the mere approachability (i.e. being neither trivial nor hopelessly difficult to analyze)? 

 A: See Chaitin's paper An Algebraic Equation for the Halting Probability, which is on essentially how to write a Lisp interpreter as a huge Diophantine system. Certainly interesting, but a bit beyond what this MO question is asking for. :)
A: In part the motivation comes from applications, such as physics. Some of the more recent interest in Calabi-Yau varieties e.g. was triggered by the discovery of mirror symmetry by string theorists. Certain classes of families of diophantine polynomials describe simple types of Calabi-Yau spaces in toric varieties and provide a fairly large number of quite different types of diophantine equations. During the first post-mirror symmetry decade this interest came mostly from classical algebraic geometers, but over the past few years  some number theorists have become interested in Calabi-Yau spaces as well. The question of automorphy of Calabi-Yau type motives is an example of a concrete problem. This problem is of interest already in dimension two, i.e. for families of K3 surfaces described e.g. by hypersurfaces in weighted projective spaces or toric varieties. In the case of CY threefolds this problem has played an important role in the work of Clozel, Harris, 
Shepherd-Barron, Taylor on the Sato-Tate conjecture. This involves a 1-parameter family of quintics in projective 4-space 
${\mathbb P}^4$.
A: My point of view is that one is really interested in the rational points of a particular variety (or class of varieties).  The diophantine equation ``comes along for the ride,'' so to speak.  For example, one is interested in questions like: does a variety have a rational point? Are the points dense for various topologies (Zariski, adelic)? Does the class of varieties our particular example comes from satisfy the Hasse principle?
It turns out that the answers to these questions tend to be invariant under birational transformations: e.g. if $k$ is a number field, the Lang-Nishimura lemma says that if $X' \to X$ is a birational map between proper integral $k$-varieties then $X'$ has a smooth $k$-point if and only if $X$ has a smooth $k$-point.
This suggests that we let birational classification results help us decide which classes of varieties (and hence what kinds of diophantine equations) to study. Morally, the more geometry we know about a particular birational class, the more we'll be able to say about the arithmetic of the variety (and hence the solutions of the associated diophantine equations), which is to say I strongly agree with JSE's last paragraph.
A: As has been well told by others, there are many interesting classes of diopantine equations (norm equations, elliptic curves and abelian varieties, curves of genus > 1, S-unit equations, varieties where the rational points are always Zariski dense or are always finite, failure of the Hasse principle (or not), Lang's conjecture, etc.). However, it is a theorem of Wiles that are no more specific diophantine equations of interest. Any particular problem, say, the existence of infinitely many integral solutions to $x^3+y^3+z^3 = 3$, will only be interesting to the extent that the solution sheds light on the general arithmetic properties of surfaces. Fermat was special, for a combination of historic and aesthetic reasons. 
A: Picking up on the theme of the Hilbert problem on diophantine sets: we do know that they comprise all recursively enumerable sets. A diophantine set being only slightly more sophisticated than a given equation (at least at first sight), can we really answer the question "which recursively enumerable sets are interesting?" What that implies, really, is that if you leave the confines of traditional studies (low degree, small number of variables), and diophantine geometry (roughly, equations matching up with famous types of algebraic varieties) the only answer would be "and so what do you find interesting?" 
I actually have met a logician who claimed that the theory of large cardinals is not about bogglingly large cardinalities as such, but about certain diophantine equations. Considering the set of theorems in an axiomatic set theory as a recursively enumerable set, this perspective can hardly be refuted. If your mathematical interests are encapsulated in some formal theory, then equally there may be diophantine equations you'd regard as interesting. Of course one hardly expects to deal directly with the equations.
A: $x^2 - ny^2 = 1$ is interesting for at least two reasons:  on the one hand, $x^2 - ny^2$ is a norm from the quadratic field, so the equation has to do with the rather natural question of studying units in real quadratic fields.  On the other hand (or, really, on a different finger of the same hand) it is just what you want to study if you are interested in rational approximations to square roots of integers, which in some sense are the "simplest" irrational numbers and thus the first context in which you might think about approximating irrationals by rationals.
Similarly, the Fermat and generalized Fermat equations are quite natural in the following sense:  there is a long history of studying the interplay between addition and multiplication in integers, and in particular the additive relations between multiplicatively defined sets (primes, perfect powers, etc.)  In this context it makes sense to think about $x^n + y^n = z^n$ and things like the Goldbach conjecture.  What makes the former more natural?  In some sense, it is natural because there's an approach to it!  It turns out that the equation $x^n + y^n = z^n$ is intimately related to the geometry of $P^1$ - three points (in some sense the algebraic curve on which all others are based) and to the closely related object X(1), the moduli space of elliptic curves.
There is no hard and fast rule for "which Diophantine questions are interesting" -- but in general it is not so far off to say that the ones which are interesting are the ones where we have at least some idea how to attack them, because the reason we have some idea how to attack them is typically because they're connected to some other mathematical objects of interest. 
A: Pell equations were used in Matiyasevich's solution of the Hilbert 10th problem. They are related to continued fractions, and many other important things. But originally they, I think, were studied because they are nice looking equations. Cubic equations are related to elliptic curves. Higher genus equations are related to Diophantine approximation and many deep results in arithmetic geometry (Faltings and others). In general, Diophantine equations are considered interesting by themselves (just as physics applications), and if some new method helps solving some class of Diophantine equations, the method is automatically considered useful even though the Diophantine equations may not be useful (yet).  
A: The Greeks were interested in geometry but, at the same time, they preferred their quantities to be rational, so they naturally asked when could they construct geometric objects where the measurements were rational. Rational solutions to $x^2+y^2=z^2$ is the obvious example.
Nowadays we are still interested in geometry, albeit of a different kind. Algebraic geometers parametrize geometric objects by moduli spaces and we can still ask when such moduli spaces have points with rational coordinates. The paramount example here is of modular curves. These parametrize elliptic curves with certain properties and having (or not having) one such defined over $\mathbb{Q}$ is nice, so modular curves are interesting diophantine equations.
A: The question that was asked compares Diophantine equations to differential equations, with the famous differential equations first arising due to physical arguments before taking on a life of their own.  The interests of mathematicians long ago in simple questions about geometry or powers of numbers are what gave rise to the classical Diophantine equations.  That such equations still have interest is due to them "taking on a life of their own": connections are found with important themes of mainstream mathematics, so those old equations become good examples of advanced theories.
For example, special instances of Pell's equation $x^2-dy^2=1$ occurred in the work of Greek and Indian mathematicians thousands of years ago. One reason is related to irrationality. Since $\sqrt{2}$ is irrational, $x^2 - 2y^2$ is not $0$ when the variables are positive integers and you might ask, particularly in those old days when there was not very advanced math,  what the smallest nonzero integral value of $x^2-2y^2$ could be, and how such values occur. This leads to $x^2-2y^2 = +1$ or $-1$, and both equations have many integral solutions by a recursive method, as the Indians knew. If you look at $x^2 - 3y^2 = +1$ or $-1$ you find quickly that there's no solution when the right side is -1, so already something new happens.
Another way Pell's equation arises is through questions about polygonal numbers, which were a topic of interest long ago. (I am not going to argue that they have some over-arching signifiance today, but what do you expect people back then to have been thinking about?)  Since $$36 = 6^2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8$$ is both a square number and a triangular number, you can ask whether there are square-triangular numbers beyond 36, and finding more examples is essentially the same as solving an instance of Pell's equation.
Many geometric questions about triangles, esp. right triangles, with integral or rational side lengths lead to low-degree Diophantine equations.  The equation $a^2 + b^2 = c^2$ is too famous to say anything about. Fermat was inspired to show $x^4 + y^4 = z^2$ has no nontrivial integral solutions in order to prove no right triangle with rational side lengths can have area equal to a perfect square (you can't "square" a rational right triangle).  As an unplanned consequence of solving that problem, Fermat had shown the Fermat equation with exponent 4 has no nontrivial integral solutions (replace $z$ with $z^2$ in the previous equation). The method discovered by Fermat for this problem was his technique of infinite descent, which he was able to use successfully on other problems, including those with a more positive character (i.e., showing some equation has an integral solution, like primes $p \equiv 1 \bmod 4$ being a sum of two squares).
The link found later between Pell's equation and unit groups in quadratic rings provided a reason for number theorists to have a permanent conceptual interest in that equation.
Once we get tired of squares all the time, we might look at squares and cubes. 
The progressions of perfect squares and cubes keep interlacing and you might ask how close they can come (other than the silly case when they coincide, like with $64 = 8^2 = 4^3$). 
This leads to $y^2 = x^3 + 1$, $y^2 = x^3 - 1$, $y^2 = x^3 + 2$, and $y^2 = x^3 - 2$. Here we see a very different situation compared to Pell's equation, since these equations will have only a finite number of integral solutions; the case of $y^2 = x^3 - 2$ is a famous example Fermat used to challenge the British mathematicians. We don't know how Fermat showed the only integral solutions are $(3,5)$ and $(3,-5)$, but Euler discovered later that prime factorization in the ring $\mathbf Z[\sqrt{-2}]$ gives a natural explanation of the result.  This was one of the earliest instances of using algebraic integers to solve Diophantine equations in ordinary integers, and still provides a good example for an algebraic number theory course.  
Euler looked at $y^2 = x^3 + 1$ and $y^2 = x^3 - 1$ and found a way to apply Fermat's idea of infinite descent to show the only integral solutions are the small ones you can find by hand. In the early $20$th century Mordell pushed the method of descent further to prove the Mordell part of the Mordell--Weil theorem.  Through the influence of Weil and others, the method of descent remains an important tool, although in a language that looks nothing like what Fermat used. 
Mordell spent many years of his life studying integral solutions of the equation $y^2 = x^3 + k$, where $k$ is a fixed nonzero integer.  The equation could be justified as having interest because it's one of the simplest examples of an elliptic curve, but it's important for a better reason.  The $abc$-conjecture, which has connections to many other problems, does not at first look like it is about Mordell's equation. However, the $abc$ conjecture turns out to be equivalent to specific upper bounds on relatively prime   integral solutions $(x,y)$ to Mordell's equation $y^2 = x^3 + k$ in terms of the parameter $k$.  So, as Barry Mazur once remarked, the Mordell equation is a far more central topic to all of number theory than its rather special appearance suggests.
A: Added 22, October:
While reflecting on this question and the subsequent discussion, I fell again into a
bit of a sermonizing mood. I hope  I will be forgiven for inserting two words of
caution.


*

*There is a difference, probably significant, between lack of interest
and aggressive lack of interest. Most people will know what I mean by this,
I think. The latter is probably best avoided. Now, I wouldn't be too strong about
doing away with
it altogether since some people do seem to derive substantial creative
energy by being against something, both in mathematics and in the world at large.
But for most of us,
the less contemplative version of disinterest is, I believe, more destructive than
constructive.

*Since I've already expounded on the mainstream view,
it might be alright to vary on it a bit.
It's good to educate oneself about fashion and probably sensible to
follow it  a fair amount.  But then there are the true cliches about
independent thinking.
I'm old enough to have witnessed first-hand the fervor surrounding the
wonderful proof of
FLT. At this point, I'm sure many of us can recite by heart all the reasons it is more important
than Goldbach, for example, and the significance of the relation to modularity, etc.
This may indeed be a reasonable point of view.
However, to be honest, I rarely got the impression then that the young expert in the common room was
doing more than just that: reciting the viewpoint.
I suppose  I'm just repeating the platitude that fashion might be quite
sensible, but slavish adherence to it is not. So if you feel strongly about
some specific Diophantine equation that doesn't quite fit the standing paradigms,
my own advice is to to think about it frequently enough to see if some real ideas develop. I hope I'm not misrepresenting him, but Swinnerton-Dyer once claimed that very few people were interested in L-functions around the time he was first experimenting with points on elliptic curves. Even now, he will speak with considerable passion about a single equation, or at least, about a single special family.
(Nonetheless, I hope  these paragraphs don't strike you as regurgitation of some superficial
faith in 'diversity'. I have some of that as well, but some mathematics  is clearly better than others.)
Regarding Goldbach, I have the curious impression that it's about to gain substantially in respectability,
especially with the remarkable ascent of additive number theory related to the work
of Gowers, Green-Tao, et. al. I  try to think about it myself
every now and then,  partly because of the influence of Shinichi Mochizuki,
who insists on viewing the connection between additive and multiplicative structures in arithmetic
through the prism of non-abelian fundamental groups.

Oops! I'd better clarify right away that my remarks on Fermat and Goldbach are meant in no way as criticism of Jordan's nice answer.

Original answer:
A proper answer might require at least an essay, but here is an abridged attempt.
Two classes of equations have already been discussed in the other answers:
(1) Some equations are 'just interesting' for their special or exotic properties. I quite
 like the classical mathematics generated by the equation $$x^3+y^5=z^2$$ mentioned in Mike Bennett's comment. Smooth cubic surfaces like
  $$x^3+y^3+z^3+w^3=0$$
  are also nice with their twenty-seven lines that eventually do influence their arithmetic. Fermat equations might be appreciated for their similarly high degree of symmetry that  induces the complex multiplication on their Jacobians. By the way, regardless of their age, I find Calabi-yau varieties quite fascinating myself, since the interplay between Hodge-theoretic and Diophantine properties
   is a subtle phenomenon deserving of study.
There is obviously no objective criterion being offered  here, but still an equation may appear as especially interesting in the same way certain spaces are interesting or some animals are interesting. 
   They excite a certain desire to know about them in considerable detail. Investigation with affection is usually richly rewarded in these cases.
(2) Equations that come up while studying some other problem. Pell's equations were mentioned above, and one could consider other norm equations while studying number fields. Keith mentioned also the  relation between the far-reaching ABC conjecture and Mordell's equation. One other nice class of elliptic curves of this nature are $$y^2=x^3-n^2x,$$ which were famously connected to the congruence number problem on the area of right angle triangles with rational sides and areas.  A spectacular example in the ABC vein is Mazur's study of points on modular curves that gave rise to uniform bounds on the torsion subgroup of all elliptic curves over $\mathbb{Q}$. And then, integral points on Siegel moduli spaces were bounded by Faltings in his proof of the Mordell conjecture. In short, one can even apply one kind of equation to the study of another (family).
In any case, for this class, one presumes the equations are as interesting as the motivating problem.
However, the perspective I actually wished to mention sidesteps the question somewhat. The view is perhaps the most mainstream and reactionary possible in this context, but closest to mathematical practice as I see it. It says most equations have or lack interest not in and of themselves. Rather, the main issue is the questions we ask about them. I will remind you of three examples:
(i) Consider the various conjectures on $L$-functions. Say the conjectures of Birch and Swinnerton-Dyer. To oversimplify the case a bit, suppose you could prove the conjecture in full up to the last detail for the single elliptic curve
$$498208y^2=x^3+309208472x^2+1204948278x+3920984$$
or with some other choice of coefficients  as random as you want. There will be little disagreement that this would be highly interesting.
In case you think that elliptic curves are already deserving of
special attention, choose a random collection of homogeneous equations
$$f_1=0, f_2=0, \ldots, f_n=0$$
in $m$ variables. Most of the time, they will define a smooth projective variety $X$. Bloch and Beilinson have conjectured that the order of vanishing of $L(H^{2i-1}(X),s)$ at $s=i$ is equal to the rank of the Chow group of algebraic cycles of codimension $i$ homologically equivalent to zero. Being able to prove that statement for any given $X$ chosen randomly would be highly interesting. Of course because one expects this to be so difficult, people concentrate rather on special $X$'s. [In case you are wondering about their relevance, algebraic cycles on $X$ should rightly be thought of as 'generalized solutions'.]
(ii) Continuing with the same notation, suppose $X$ happens to be a Fano variety, which will often happen if
the degrees of the polynomials add up to something rather small compared to the number of variables. In that case,
Manin has conjectured that the rational solutions in some fixed number field $F$ (depending on the equations) will be Zariski dense. That is, this is a set of equations with very simple geometry, because of which  it should be consistently easy to find many, many solutions, as soon as some obvious obstruction is overcome.
Once again, you are free to attempt this after choosing the $f_i$'s in as arbitrary and as unaesthetic a manner
as possible. As with the Beilinson-Bloch conjecture, the more random your choice is, the more impressive your
result will be, in some sense.
(iii) Close to my own heart is the effective Mordell conjecture, which asks for an algorithm to find all rational
solutions to a generic equation
$$f(x,y)=0$$
with degree at least 4. As a consequence of the fact that such an algorithm is unknown, it becomes of considerable interest to
be able to list full solution sets in any given case. Sometimes, it's easy to show that there are none, such as
$$x^4+y^4=-1,$$
just to be absurdly simple. However, once such silly reasons for triviality are excluded, for example, if you happen to notice already one
 solution, it is notoriously difficult to list the whole solution set. Here is an example due to Bjorn Poonen:
$$y^2 = x^6 - 2x^4 + 2x^3 + 5x^2 + 2x + 1.$$
You will easily see the solutions $(0,\pm 1)$. However, it requires quite a bit of technology to show that
$$(0,\pm 1), (-1,\pm 1), (1,\pm 3)$$
is the full solution set. You can see that this particular equation does seem pretty random. On other hand, because of an enduring
focus on the difficult and structurally demanding question of an algorithm, any example of this sort generates quite a bit of interest.
Many people have suggested that the variety of techniques that come up in attacking a problem are as important as the problem
itself. There is something to this, in as much as we would like the problem to tell us as much as possible about
the mathematical landscape in general, which is, after all, the ultimate object of our investigation. On the other hand,
once certain overarching
 questions have already been established as powerful probes for this process, being able resolve them for any specific 
 object is interesting regardless of how pretty or ugly someone may find the object on its own. Obviously, this
 is the raison d'etre for good conjectures.

Added 26, October:
Eventually, I stumbled on to the 'box equation' I referred to in the comments. It is
$$a_1^2+a_2^2=b_3^2;$$
$$a_1^2+a_3^2=b_2^2;$$
$$a_2^2+a_3^2=b_1^2;$$
$$a_1^2+a_2^2+a_3^2=c^2;$$
defining a surface in $\mathbb{P}^6$. A rational solution with $a_1a_2a_3\neq 0$ corresponds to a 'rational box' having all edges, face diagonals, and long diagonal rational. Apparently, the existence of such a thing is still unknown. There is a nice discussion in this 
paper of Stoll and Testa. Of course, you have to decide for yourself if it's interesting. The flavor of it is somewhat reminiscent of the congruent number problem, and I think that was why it caught my attention. That is, given my own bias, I had to consider for a minute or two if  there were a sneaky  connection to a 'conceptually sophisticated' problem. Stoll and Testa relate it, in fact, to the Bombieri-Lang conjecture.
A: This certainly isn't something I thought a lot about, but there has definitely been interest about "Generalized Fermat Equations" (like the one you listed). Here's a quick link that I found googling it that has more or less the theorems I remembered:
http://www.claymath.org/publications/Arithmetic_Geometry/Chapdelaine.pdf
