When I updated my iPad 2 to iOS 7 I had a similar problem. The installation of iOS 7 went smooth, but music on the iPad would not play so I restored and re-loaded my backup. Then iTunes told me it didn't have enough room to sync the selected playlists I'd chosen, even though they previously fit with >1 GB to spare.
It ended up that >1800 past iTunes purchases made directly from the iPad were trying to sync, and they could not be removed even though I had changed my settings to not sync any music. To clear this up I ended up doing the following:
1- Changed to manual sync mode.
2- While connected to iTunes, selected all 1800+ stubborn songs and deleted them.
3- With no music loaded I saved my backup to iTunes.
4- Restored again and loaded the backup from prior step.
5- After confirming no music was loaded, changed back to auto-sync to my selected playlists.
6- Several hours later, all music I wanted was loaded, and plays correctly.
A royal pain, no doubt, but at least I didn't have to restore as new without a backup (which would have required a lot more work).
My iPhone 5 had no such problems with the iOS 7 update, which is set to sync the same playlists as my iPad, so it is either a random problem, or one that is specific to certain hardware platforms.