Abstract homotopy theory allows one to use the tools of homotopy theory (e.g. inverting weak equivalences, computing homotopy colimits, doing Bousfield localization, taking fibrant and cofibrant replacements, etc) in many different settings. In some of these settings (e.g. homological algebra), it's not a huge surprise that you can do so. But settings include

• Representation theory (via the stable module category)
• Algebraic geometry (via motivic homotopy theory)
• Graph theory (via work of Bissen and Tsemo)
• Category theory (via work of Rezk, among others)
• Universal algebra (via colored operads and PROPs)
• Mathematical physics (via TQFTs)
• Dynamical systems (via Gaucher's work on flows)
• Computer science (via work of David Spivak, among others)

All these settings form model categories, so you can carry out your favorite constructions from homotopy theory. In each setting, the methods of abstract homotopy theory have been used to prove new theorems. I think as a field we need to market ourselves a bit better, and show the people in these areas that we're not trying to force them to use our language, but we are offering them some tools that they might find useful.

When one works in the abstract setting, it's common to try to specialize to get results in some or all of these areas. Most abstract homotopy theorists do still care about topological spaces and spectra, so will include examples specializing to those settings in their papers. In terms of the questions abstract homotopy theory asks, it seems they come in several flavors:

1. Finding new examples to add to the list above, or proving examples encode the same homotopy theory (if they are model categories, this is asking for a Quillen equivalence).
2. Taking problems in areas on that list and trying to solve them using the abstract methods. For example: the cobordism hypothesis, Milnor's conjecture, Vandiver's conjecture, Happel's book on representation theory.
3. Computing homotopy groups, algebraic K-theory groups, Tate cohomology, etc. Here, the abstract setting is what allows you to bring over tools of spectral sequences and other transfinite computational processes.
4. Developing additional unifying structure, e.g. finding the right way to encode some definition or theorem, so that it recovers seemingly different things on items in the list above, proving that they are in fact special cases of the same general concept.

In terms of the difference between model categories and $\infty$-categories, it seems the field has remained fairly civil about the fact that there are two settings in which you can "do abstract homotopy theory." Each has strengths and weaknesses. I use model categories because it feels closer to my intuition from spaces, because all the examples I care about are model categories, because I have more tools to use in that setting (e.g. fibrant/cofibrant replacement), and because I've always been able to make things work in that setting. If I was ever trying to prove something and got sidelined by technical details I might switch to $\infty$-categories if that would make life easier.