Philip gave a nice answer but let me add some points to it.
The thing is that while IMT and DST were far apart back in the days, over the years it became clear that part of DST and IMT deal with exactly the same problem.
The above sentence may not be completely objective, but at any rate, that seems to be the point of view of some people who do inner model theory (including mine). The main goal of IMT is to construct models that are correct about the universe. How you measure this correctness? Well, there is Levy hierarchy and there is a more refined and more useful version of it, namely the Wadge hierarchy. So to make the question more precise, what we do in inner model theory is that we try to construct and analyze canonical models called mice that capture the levels of the Wadge hierarchy. Back in the days, it wasn't so clear that there is such a deep relationship between Wadge hierarchy (or their more refined forms, the Universally Baire sets, or homogeneously Suslin sets) and the large cardinal hierarchy. Now, the connection is much more clear and transparent.
DST is really inevitable. The technical problem that people doing IMT are trying to solve is the construction of omega_1+1 $\omega_1+1$ iteration strategies. These iteration strategies allow one to build trees of height omega_1 $\omega_1$ and then it is guaranteed that any such tree must have a branch. But wait a minute, in ZFC alone, there are trees of height omega_1 $\omega_1$ with no branchbranches. So how are we going to make sure that these strategies will build only trees of height omega_1 $\omega_1$ for which there are branches. The answer must be that the strategy has various DST like properties, like it is Universally Baire or hom Suslin and etc (it is a nice exercise to show that such strategies are indeed nice).
At any rate, when doing IMT, running to DST seems to be inevitable. It is probably not so true if you are doing DST (by DST we mean Moschovakis' book not the directions it went since 90s). But still there are many theorems you can prove under AD using IMT and no proof avoiding IMT is known. For instance, every regular cardinal below Theta is a measurable cardinal, a fact proven using IMT and no proof avoiding IMT is known.
The reason that the subjects are so close, though, is just that they study the same object (namely the Wadge hierarchy) from different point of views. DST does it using recursion theoretic methods (coding lemma, pointclass arguments and etc) and IMT does it by translating the Wadge hierarchy into hierarchy of iteration strategies, which is what IMT studies.
