You can't listen to full songs on the service unless you sign in with your Spotify or Rdio account so that limits the service quite a lot. Like me, if you don't use Spotify or Rdio Twitter Music only lets you listen to 30 seconds of a song which is pretty pointless.
This makes the service better for discovering music rather than listening to it.
The main page of the app and website show what are the most popular songs at the moment, it allows you to follow any singer on the service really easily by just clicking the Twitter button and you can Tweet about what song you are listening to (which I can imagine getting quite annoying for your followers if you are always doing that.)
In conclusion, Twitter Music is extremely limited as only gives you a small handful of unique features. I personally will probably not be returning to the app or website again until some radical changes are made.