Sony will be able to run music in background and such but they have never showed anything to the degree of snapping apps.
As long as you subscribe to their Music Unlimited service.
Will I be able to play background music while I play PS4 games?
Yes, the Music Unlimited streaming service will allow you to listen to music while you play PS4 games. A subscription to Music Unlimited is required. Get full details here.
Does PS4 include a music visualizer feature?
No.
Will I be able to play MP3s on PS4?
No.
http://blog.us.playstation.com/2013/10/30/ps4-the-ultimate-faq-north-america/
Seems pretty sh!tty to me. Anyone know what MS' stance is on background music?