For starters, the app design is very impressive. It's sleek, simple, and it focuses on music discovery more than any other app. You're able to tell Beats Music what mood you're in and it will instantly create a curated stream that (hopefully) matches your mood. It's heavily based on curation, and the more you use it, the more it can learn about you. It's your own personal DJ, in a sense.
There's also the typical stuff: being able to pick and listen to pretty much any track in the known world, and the ability to build your own playlists. For those who pony up $10/month, you'll be able to save albums and playlists for offline listening, as well as no ads and unlimited album/song downloading.
Sadly for MOG users, that service will be shut down on April 15th. Some time ago, Beats purchased MOG, and this is the eventual conclusion. MOG users probably won't be too thrilled with the news, but such is life in the startup world. MOG billing will cease as well, and while MOG users won't necessarily have to switch to Beats, the company plans on providing tools to existing users to make that transition a little easier.