(Why would the servers be required anyways? To stream content? Couldn't content be downloaded on first run (which means you only need to fund a server&pipe big enough for new purchases))
What? Are you being intentionally ridiculous. They license songs. I don't know the terms of these specific licenses but the way they work in general is that they can license the game for a specific use and/or term. I'm not a lawyer so I don't know specific legal terms off the top of my head but there's a different license for the PlayStation version of the game where they are giving the user a copy of the song to have and to hold and one where they would stream, which I assume the iOS version is actually doing. Again, not a lawyer so I don't know the specifics but you will often see something like a TV show having broadcast rights to use a song in their show but they (especially for older shows) may have failed to procure the appropriate rights to use the song on the DVD version.
Now, I don't know too much about the Rock Band app so I'm sure someone more knowledgeable about it is likely to come through and correct me but it sounds like the songs are streamed for the iPhone. This wouldn't be as crazy as it sounds since phones still have relatively limited space and applications with smaller footprints are thus easier to manage.
When you buy Guitar Hero or Rock Band on the playstation, the disc is yours to keep and play forever (I assume). I'd expect the same from an iOS app.
Especially since the .ipa download is 160 MB.
3.3.16 Any master recordings and musical compositions embodied in Your Application must be wholly-owned by You or licensed to You on a fully paid-up basis and in a manner that will not require the payment of any fees, royalties and/or sums by Apple to You or any third party. In addition, if Your Application will be distributed outside of the United States, any master recordings and musical compositions embodied in Your Application (a) must not fall within the repertoire of any mechanical or performing/communication rights collecting or licensing organization now or in the future and (b) if licensed, must be exclusively licensed to You for Your Application by each applicable copyright owner.
Edit: And will this have consequences for any other EA offerings?
Probably depends on licensing terms. I can absolutely see record companies licensing under terms that don't allow end-users to save the music locally.