I don't think you can say that detecting the unknot and detecting knot genus have been long understood just because there were algorithms for them. You want effective algorithms, or ways of computing infinite families of examples.

Aside from that, Ozsvath and Szabo had early applications to questions about which three manifolds can be obtained from which other manifolds by what kinds of surgeries. The Ozsvath-Szabo contact invariant has a number of applications, for example to understanding symplectic fillings. There is much more, too much for me to completely follow, so that anything I say would be very incomplete.

In addition, most of the applications of Seiberg-Witten theory (e.g. the Thom Conjecture, distinguishing various non-diffeomorphic but homeomorphic smooth manifolds) can be re-proved using the Ozsvath-Szabo theory. Since the two theories are known to be equivalent, it is a matter of taste whether you want to regard these as applications of Seiberg-Witten or Ozsvath-Szabo.