How do I find out what apps are in iTunes and not on my iPhone, and vice versa?
For apps that are in your library [associated to an account] but not on your phone, go to iOS App Store > Updates > Purchased, then tap the "Not on this Phone" tab.

For app that are on your phone, but not in your library. Connect your phone to you computer, load iTunes, select the phone from the connected devices list and go to the apps section. Each app that is installed on your phone will be missing the Install/update button next to it.