For everyone who's having a problem downloading new apps in iOS 6, or are missing apps that didn't transfer over from iTunes, here's a super quick solution.
If you have any apps that are currently stalled, you are going to want to stop and delete those and then just re-download them after the fix below.
This seems to be more of a problem for older devices like the iPhone 4 and iPad 2, but it should work for newer devices nonetheless.
- Sign out of your Apple Account.
- Eject your iPhone and Quit iTunes.
- Go to Settings > General > Reset and Reset Network settings.
- Restart your iPhone (hold the Sleep button and Home button).
- Log back into your iTunes account and start your downloads again.
Boom, pow, done.