What are the advantages of the more abstract approaches to nonstandard analysis? This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the abstract approaches to NSA (e.g., via the compactness theorem), as compared to the more concrete approach using ultrapowers? One can name generic reasons such as naturality, functoriality, categoricity, etc., but I am hoping for a concrete illustration of why a more abstract approach may be advantageous for understanding NSA concepts and/or proving theorems.
Note 1. One of the existing answers provided a bit of information about advantages of the more abstract approach in terms of saturation. I would appreciate an elaboration of this if possible, in terms of a concrete application of saturation.
Note 2. These issues are explored in more detail in this 2017 publication in Real Analysis Exchange.
 A: For most purposes, I think the premise is wrong: in many situations ultraproducts simply are the preferable approach, so I'll try to discuss some of the purposes which are exceptions.  @cody has already mentioned the issue of axiomatics, which is a reason to avoid ultraproducts.
Another reason to avoid ultraproducts is that nonstandard analysis actually handles iterated infinitesimals more easily.  It's common in nonstandard arguments to take a nonstandard number H, and then take a second number which is not only nonstandard, but nonstandard (i.e. sufficiently big) relative to H.  I've never seen an argument like this done in the ultraproduct version.  Though of course there's no formal obstacle, for arguments like these the concreteness of the ultraproduct becomes a disadvantage, and getting it right would require either taking multiple ultraproducts or some fiddling with the underlying construction to choose the right elements.  The abstract approach, on the other hand, is very natural: having gotten used to working in NSA, doing it repeatedly adds no new complications.*
This is really a special case of the issue of saturation.  The point of nonstandard analysis/ultraproducts is that certain infinite intersections of sets are guaranteed to be non-empty.  There's a lot of room to detail exactly which infinite intersections this applies to.  Ultraproducts are useful in part because they give a definitive answer which is the right one for many purposes.
But the abstract approach is more flexible: it can consider both less saturated versions (structures which are nonstandard but don't have "$\aleph_1$-saturation"), and also more saturated versions (like the double saturation implied by the item above, or by demanding $\kappa$-saturation for larger cardinals---that is, considering intersections of more than just countable sets).
*: In fact, I'm not certain one can do the analogous arguments using only a single ultraproduct---one might have to take the ultraproduct twice to get the right result.
A: You can quote me on this if you like.
This is such an old issue that I am surprised it is still up for discussion.   I know you like ultra-powers, etc,  but I have always thought they were wrong-headed;  it was Luxemburg who made them popular for NSA originally.
The point is that you actually really NEED the transfer theorems; so you essentially need the logical apparatus; in most cases the compactness theorem and some form of saturation.    Occasionally a type omitting argument could be used but that is rare.
The reason is that you are postulating that "all of mathematics" carries over to the
non-standard model and in fact, that is the underlying intuition in the applications.
I also think that the "more concreteness" of the ultraproduct construction is just an illusion.     You can not answer if the integer produced by (1,2,3,4,5, ...) is even or odd
since it depends on the ultrafilter completion of e.g. any non-principal filter.
Moreover, the basic tools in almost all the proofs is dealing with the seam between
"standard" and "non-standard" elements; which really depends on a feel for expressibility in the language being used.  Nelson's approach was to try to help people manage without the "feel".  That's a matter of taste.
Furthermore, I would differ with your point that NSA is to formalize classical mathematicians arguments.  That is "cool" and Robinson greatly enjoyed it, but really he also was sure that it is a great way to prove new theorems.    Since people sort of like to
hear about the theorems consequence in "classical" settings; then one often had to find
equivalence theorems (this was clear in the work on brownian motion etc) and in my work on inverse limits of finite groups  (by the uniqueness of inverse limits, the non-standard finite groups are essentially equivalent to inverse limits of systems of finite groups and a lot of the very early work of Lubotzky on profinite groups can be easily carried out in this setting)  but actually
a more extreme position can be seen when you just take the position that the "standard"
results are just one specific implementation and one could develop mathematics happily without them.    The argument that the non-standard models are not "unique" is just a habit and not important.   
Robinson himself put this viewpoint forward very nicely in his Brouwer medal address  "Standard and NonStandard Number Systems"
Best regards
Larry Manevitz
manevitz@cs.haifa.ac.il
Regarding the use of limit ultrapowers etc; one needs to be careful there.  I once proved there was no measureable cardinal by taking such a large ultrapower; but actually all the
standard sets dont grow in that context :).
A: IST easily maintains many levels of standardness (sort of). Take an inf. large integer $H_0$. The family $F_0$ of all values $f(H_0)$, $f:\mathbb N\to \mathbb N$ being standard, is bounded in (the nonstandard) $\mathbb N$, so there is an inf. large integer $H_1$ bigger than any number in $F_0$. Replace $H_0$ by $H_1$ in this argument, getting $H_2$, and so on. Then each $H_{k+1}$ behaves in many ways like an inf. large integer wrt $H_k$. My book with Reeken considers relative standardness in detail.
A: To my way of thinking, there are at least three distinct
perspectives one can naturally take on when undertaking work in
nonstandard analysis. In addition, each of these perspectives can
be varied on two other dimensions, independently. Those dimensions are, first,
the order of nonstandardness (whether one
wants nonstandardness only for objects, or also for functions and
predicates, or also for sets of functions and sets of those and so
on), and second, how many levels of
standardness and nonstandardness one desires. Let me describe the three
views I have in mind, and give them names, and then discuss how
they relate to one another.
Classical model-construction perspective. In this approach,
one thinks of the nonstandard universe as the result of an
explicit construction, such as an ultrapower construction. In the
most basic instance, one has the standard real field structure
$\newcommand\R{\mathbb{R}}\newcommand\Z{\mathbb{Z}}\langle\R,+,\cdot,0,1,\Z\rangle$,
and you perform the ultrapower construction with respect to an
fixed ultrafilter on the natural numbers (or on some other set, if
this was desirable). In time, one is led to want more structure in
the pre-ultrapower model, so as to be able to express more ideas,
which will each have nonstandard counterparts. And so very soon
one will have constants for every real, a predicate for the
integers $\Z$, or indeed for every subset of $\mathbb{R}$ and a
function symbol for every function on the reals, and so on. Before
long, one wants nonstandard analogues of the power set $P(\R)$ and
higher iterates. In the end, what one realizes is that one might
as well take the ultrapower of the entire set-theoretic universe
$V\to V^{\omega}/U$, which amounts to doing nonstandard analysis
with second-order logic, third-order, $\alpha$-order for every
ordinal $\alpha$. One then has the copy of the standard universe
$V$ inside the nonstandard realm $V^*$, which one analyzes and
understands by means of the ultrapower construction itself.
Some applications of nonstandard analysis have required one to
take not just a single ultrapower, but an iterated ultrapower
construction along a linear order. Such an ultrapower construction
gives rise to many levels of nonstandardness, and this is
sometimes useful. Ultimately, as one adds additional construction methods, this amounts as Terry Tao mentioned to just adopting all of model theory as one toolkit. One will want to employ advanced saturation properties or embeddings or the standard system and so on. There is a very well-developed theory of models of arithmetic that uses quite advanced methods. 
To give a sample consequence of saturation: every infinite graph, no matter how large, arises as an induced subgraph of a nonstandard-finite graph in any sufficiently saturated model of nonstandard analysis. This often allows you to undertake finitary constructions with infinite graphs, modulo the move to a nonstandard context. 
Standard Axiomatic approach. Most applications of nonstandard
analysis, however, do not rely on the details of the ultrapower or
iterated ultrapower constructions, and so it is often thought
worthwhile to isolate the general principles that make the
nonstandard arguments succeed. Thus, one writes down the axioms of
the situation. In the basic case, one has the standard structure
$\R$ and so on, perhaps with constants for every real (and for all
subsets and functions in the higher-order cases), with a map to
the nonstandard structure $\R^*$, so that every real number $a$ has
its nonstandard version $a^*$ and every function $f$ on the reals
has its nonstandard version $f^*$. Typically, the main axioms
would include the transfer principle, which asserts that any
property expressible in the language of the original structure
holds in the standard universe just in case it holds of the
nonstandard analogues of those objects in the nonstandard realm.
The transfer principle amounts precisely to the elementarity of
the map $a\mapsto a^*$ from standard objects to their nonstandard
analogues. One often also wants a saturation principle,
expressing that any sufficiently realizable type is actually
realized in the nonstandard model, and this just axiomatizes the
saturation properties of the ultrapower. Sometimes one wants more saturation than one would get from an ultrapower on the natural numbers, but one can still achieve this by larger ultrapowers or other model-theoretic methods.
Essentially the same axiomatic approach works with the high-order
approach, where one has nonstandard version of every set-theoretic
object, and a map $V\to V^*$, with nonstandard structures of any
order.
And similarly, one can axiomatize the features one wants to use in
the iterated ultrapower case, with various levels of standardness.
As with most mathematical situations where one has a construction
approach and an axiomatic approach, it is usually thought to be
better to argue from the axioms, when possible, than to use
details of the construction. And most applications of nonstandard
analysis that I have seen can be undertaken using only the usual
nonstandard axioms.
Nonstandard Axiomatic approach. This is a more radical
perspective, which makes a foundational change in how one thinks
about one's mathematical ontology. Namely, rather than thinking of
the standard structures as having analogues inside a nonstandard
world, one essentially thinks of the nonstandard world as the real
world, with "standardness" structures picking out parts of it. So
one has the real numbers including both infinite and infinitesimal
reals, and one can say when two finite real numbers have the same
standard part and so on. With this perspective, we think of the
real real numbers as what on the other perspective would be the
nonstandard reals, and then we have a predicate on that, which
amounts to the range of the star map in the other approach. So
some real numbers are standard, and some functions are standard
and so on.
One sometimes sees this kind of perspective used in arguments of
finite combinatorics, where one casually considers the case of an
infinite integer or an infinitesimal rational. (I have a colleague
who sometimes talks this way.) That kind of talk may seem alien
for someone not used to the perspective, but for those that adopt
the perspective it is very useful. In a sense, one goes whole-hog
into the nonstandard realm.
More extreme versions of this idea adopt many levels of
standardness and nonstandardness, extended to all orders. Karel Hrbáček has a very well-developed theory like this for nonstandard
set theory, with an infinitely deep hierarchy of levels of
standardness. He spoke on this last year at the CUNY
set theory seminar, and I refer you to his articles on this topic.
In Karel's system, one doesn't start with a standard universe and
go up to the nonstandard universe, but rather, one starts with the
full universe (which is fundamentally nonstandard) and goes down
to deeper and deeper levels of standardness. Every model of ZFC,
he proved, is the standard universe inside another model of the
nonstandard set theories he considers.
Ultimately, my view is that the choice between the perspectives I
mentioned is a matter of taste, and that in principle any
achievement of nonstandard analysis that can be undertaken with
one of the perspectives has natural analogues that can be
expressed with the other perspectives.
A: I feel like this is an instance of a larger question:

When might it be nice to work with an axiomatic description of a theory rather than an explicit construction?

This comes up all the time, e.g. in topology, cohomology, algebra (e.g. abstract groups rather than permutation groups), and more recently with homotopy type theory.
Some possible answers to this question:

*

*Working with axioms provides the right level of abstraction: proofs often become much easier since you're left with only the essential facts, rather than the forest of theorems being obscured by the trees of the particular construction.
In NSA, you're trying to justify the algebraic manipulations done by Newton or Euler on paper: but these are naturally stated in the abstract language of calculus, and involving ultrafilters just complicates matters. Of course you can just re-derive all the required inference rules, but the point is that the logical rules themselves are a useful conceptual framework.


*Having an axiomatic framework opens the door to unexpected realizations of a concept. This is of course easy to see for a concept with many different realizations like groups, where things as disparate as field automorphisms, toplogical braids and integers all have a group structure, but can be more surprising for things like, e.g. cohomology or real closed fields, where there are only few immediately apparent models. It's probably useful to note that the concept of group itself has turned out to be much more general than envisioned originally. Once you have an axiom system, it becomes very easy to try to find "non-standard" models by various techniques (including ultrafilters, of course), or even dropping axioms, as in the famous "extra-ordinary" cohomology theories. One might even suggest that NSA itself is the result of finding non-standard models of an axiomatic theory of real numbers. It's not surprising therefore that people working with NSA might be more amenable to the advantages of the axiomatic approach.
