I have the same problem with the apps. I can't find the way to make them work on my IPhone 4s.
I have also tried:
Restoring several times to factory settings and back to different back-ups
Plugging the USB cable in a different port,
Removed data (pictures and videos) to allow more memory space,
Uninstalling and reinstalling ITunes,
Deauthorize and reauthorize my computer
When my phone is plugged in and it's in sync mode, the apps tab on the ITunes is grayed out (meaning that is not getting included). We have two old 3G iPhone and an IPad that sync from the same computer and they work fine.
Please help if you have had any success!