When I took an introductory model theory course a few years ago, the instructor did not talk about ultrapowers, which seems fair to me, given contemporary model theorists' relative indifference to ultraproducts in general.
On the other hand, ultrapowers ares still sometimes used, e.g., to introduce to students applications of model theory, such as infinitesimals or modal logic (for the latter, see, e.g., Blackburn, de Rijke, Venema's Modal Logic). As far as I know, for these purposes you don't really need ultrapowers; you just need sufficiently saturated elementary extensions, which can easily be obtained by repeated use of compactness.
(A seemingly more recent applications are in operator algebra, but I'm not sure if you really need ultrapowers as opposed to any sufficiently saturated elementary extensions.)
I would imagine that whatever property you want in your elementary extension (e.g., $\kappa$-saturated or omitting a certain type), it is impossible or harder to achieve that with ultrapowers since the latter involves intricate infinite combinatorics.
So my question is: are there situations in which ultarpowers are more desirable than other kind of elementary extensions?
(Note: I understand that there are mainly set-theoretic/combinatorial interests in ultrapowers. I also vaguely understand the uses of ultraproducts; my question is specifically in regard to ultrapowers.)