One thing is that it's very irritating to browse my non-Apple Music music now. I used to have a tab for artists, a tab for albums, and a tab for songs. Now I have one tab for my entire music collection.

With the multiple tab experience, you could browse into something on one tab, switch to another tab, and switch back, to see what you had previously browsed to (this is inherently how UITabBarController works). You can't anymore, you have to go back to the root to switch to browsing artists from albums, etc.

Another thing at all of the services completely fail at is disabling online music (iTunes Match, things not "available offline" on Spotify) when you have no cell service (i.e. on the train, when you're likely to be listening to music on your cell phone!). It's impossible to listen to streaming music at that time, so why even show it?