Winamp = a free audio and video player which comes with a lot of other services.
Windows Media Player = I can't imagine that you don't know what it is, but I'll tell you anyway, so: Windows Media Player is also an audio and video player which comes integrated with Windows (almost all versions).
See right down 5 point that Windows Media Player doesn't have, instead of Winamp:
1. Winamp has SHOUTCast Radio for free, actually this means that you have free radio on your computer using just the Internet connection.
2. Winamp has a few themes for personalize it.
3. Winamp offers you an integrated Music Library with a lot of songs.
4. Winamp gives to you the tops of the day in any idea and media subject.
5. You can play AVI files with Winamp.
Read now what W.M.P. has and Winamp doesn't:
1. Integrated CD burner.
2. Windows Media Player starts faster than any other player.
3. It has the best compatibility with dedicated buttons of computers (PLAY, PAUSE, STOP etc.).
4. It's already installed on ALL the computers operated with Windows and you don't need Internet connection.
Anyway, you can install Winamp, even just for see how it works, because as I already said it's free.
I'm pretty satisfied with WMP and I use Winamp just for radio. For AVI files I use BSPlayer, but Winamp works the same well. The biggest minus that Winamp has is that starts very hard and that's why I don't like it so much.