Podcast apps, good ones, have many features you don't need in a simple app like a winamp or vlc.
1) An online catalog of series that is searchable (probably both by name and topic and participants).
2) That catalog must contain metadata both at a series level and an episode level.
3) That online catalog must also be navigable without search, of course.
4) Ability to subscribe to any given series and for the app to have an option to pre-cache an episode for offline use.
5) Several content controls that aren't really required for music listening but are must haves for podcasts such as timed skips (30s, 60s) and playback speed adjustment (only useful in a music player where you probably want to do beat matching).
I think the biggest differentiator though is the episode/series catalog. It makes podcasts actually discoverable.