Apple Music's new design is meant to be more intuitive and more familiar. The first tab in Music will now be library, with a set of simple sections: playlists, artists, albums, and songs. There's a new Downloaded tab that's only for locally-stored music.
The For You tab now features the new Discovery Mix, with a selection of songs that are just for you, along with an integrated Connect, which has lost its own tab. Browse is curated by Apple experts with the latest music, playlists, and top charts. Music now also features support for lyrics, right in the app.
For more, be sure to check out iMore's WWDC live blog.