The significance of modularity for all Galois representations On pg. 1 of the slides of a talk, Henri Darmon wrote:

Question: What is an interesting Diophantine equation? 
A “working definition”. A Diophantine equation is interesting
  if it reveals or suggests a rich 
  underlying mathematical structure.

One can adopt the perspective that the interesting elliptic curves over $\mathbb{Q}$ are those that are modular and view the fact that all elliptic curves over $\mathbb{Q}$ are modular as being of minor significance (in the sense that if some weren't, they wouldn't be so interesting). 
I realize that my raising this perspective may come across as an affront to some of the celebrated research of recent times, and would hasten to emphasize that I'm asking my questions here in good faith with a view toward learning more.

What would we lose if we decided to
  focus only on those Galois
  representations that are attached to
  automorphic forms and ignore the
  possibility that some do not?

One thing that one would lose is Wiles' proof of Fermat's last theorem. Until recently, my attitude had been that the Frey Curve construction is a curiosity and that a "morally right" proof would come from the $abc$-conjecture applied to sufficiently large exponents together with the theory of arithmetic of cyclotomic fields to rule out the possibility of nontrivial solutions to the equations with smaller exponents. However, recently I came across slides from a talk by Minhyong Kim in which Kim wrote (pg. 29):

The idea of encoding points into
  'larger' geometry is a common one in
  Diophantine geometry, as when
  solutions
$a^n + b^n = c^n$
to the Fermat equation are encoded
  into the elliptic curves
$y^2 = x(x - a^n)(x + b^n)$.
The geometry of the path torsor
  $\pi_1(X(\mathbb{C}); b, x)$ is an
  extremely canonical version of this
  idea.

I find the idea that the Frey curve construction is canonical to be fascinating! It raises the possibility that the proof of Fermat's last theorem coming from the study of Frey equations is morally right. [Edit: As KristianJS aptly points out, I misread Kim's quote. So I'd recur to my remark above about my impression on what a "morally right" proof of Fermat's Last Theorem would look like.]
Anyway, I'd be very interested in further examples concerning the significance of all L-functions attached to Galois representations (of suitable type) being modular.
[Added: If I remember correctly, In "The Map of My Life" Shimura wrote that he was more interested in the fact that suitable cusp forms correspond to elliptic curves than in the converse. This seems relevant. However, I cannot find the quotation and may be misremembering. I would welcome a reference from anybody who remembers this.]
[Added: I just asked another question that touches on material that may provide a partial answer to this question.] 
 A: I want to make a couple of comments about the premise of this question. First, the OP asks what the consequences would be if we "focus only on those Galois representations that are attached to automorphic forms and ignore the possibility that some do not".  But of course it is certainly the case that there exist Galois representations that are not attached to automorphic forms (Galois representations that are ramified at infinitely many primes, for instance). 
Second, the OP writes about "Galois representations that are attached to automorphic forms" as though this phrase is well-defined.  But do you mean the automorphic forms that conjecturally believed to have Galois representations attached to them, or the automorphic forms that are currently known to have Galois representations attached to them?  The latter is a (growing but proper!) subset of the former; and note again that the former is not "all automorphic forms" (see e.g. the paper of Buzzard-Gee in the 2011 LMS Durham Symposium).
So, there's a conjectural map from (some) automorphic forms to (some) Galois representations, and this has been understood for decades to have significant arithmetic consequences for representations in the image of the map.  The most generous way I can think to interpret the question is "What would happen if we satisfied ourselves with statements of the form "If $\rho$ is in the image of this map in an instance where the map has been constructed...", without attempting to characterize the image in some easily checkable way?", and the answer is simply that we would lose most or all of the arithmetic consequences.
A: You would lose all the p-adic families of Galois representations, that are necessary to prove properties of the modular ones.   Hida's family was necessary for the proof ofMTT, for example.
A: Proving modularity of finite image Galois representations seems to be the most feasible way of proving the Artin conjecture. In fact, this was one of Langlands' original motivations.
A: I am not sure I understand your question. Or rather, as I understand your question, you have already answered it. Well, you asked "what would we loose if we..." and you answered "Wiles' proof of FLT". Isn't that enough? We have developed more complicated theories for much less venerable questions!
Admitting rhetorically that it isn't enough, one can develop your answer. If we want to restrict our consideration to representation that are a priori modular, 
we would loose not only Wiles' proof of FLT, but also Wiles and others's proof of the Taniyama-Shimura-Weil conjecture (and this one has nothing to do with the Frey's curve), so we would lose the
corollary that L-functions of elliptic curves satisfy a functional equation and have an analytic extension... We would then not even be able to formulate the beautiful conjecture of
Birch and Swinnerton-Dyer! We would also loose that the symmetric powers of the representation
attached to an elliptic curve are modular. So we would lose the Sato-Tate conjecture as well.
In a word, we would loose all the Galois representations coming from algebraic geometry, because we don't know a priori that they are modular/automorphic. And the fact is that those representations contain a lot of deep arithmetic and geometric (if you believe to Tate's conjecture) information.
Added: The above is about the fact that without talking of representations that are not modular/automorphic we would lose the ability even to formulate a theorem of the form "such and such representation is modular". Here's another, different, point. We also need Galois representations that are not modular, in order to prove things about representations that are modular. I am refereeing to all the arguments of p-adic families (Hida's family, 
eigencurve, eigenvarieties) who carry at each point a Galois representation which
most of the time is not modular. but we need those point to give flesh to the eigenvariety whose "skeleton" is made of the modular ones. To give an early example of those techniques,
Wiles proved that a Galois representation attached to an ordinary form is ordinary (a statement purely about modular galois representation) using Hida's families, hence non-modular representations. Nowadays those techniques are ubiquitous...  
A: Your question reminds me of a current strain of research whose starting point is Serre's conjecture, now the Khare-Wintenberger Theorem:

any continuous odd irreducible two-dimensional Galois representation over a finite field arises from a modular form

The question one might ask is then "Where are the even Galois representations?"
The answer given (mostly by F. Calegari) is that they just don't exist when you put certain additional restrictions on your Galois representation. Suppose then that you somehow have an even two-dimensional Galois representation in your hands. Well then this Galois representation is very special in some ways that might not be apparent! You can then ask: where did this representation come from? Is it modular in some non-obvious way?
So if I were to answer your question "What would we lose if we decided to focus only on those Galois representations that are attached to automorphic forms and ignore the possibility that some do not?" I would say that you lose a lot of knowledge about what makes modular Galois representations special. In your terminology, you might lose the scope of just how interesting certain representations can be!
