Several people have described a problem with App Store applications after restoring from a backup. When your applications are launched on your new iPhone 3G, they will only show the start screen and then immediate quit.
The issue may lie in the Digital Rights Management embedded in each of the applications. In order to restore functionality, you simply need to delete the applications from the iPhone itself, and then resync the applications from iTunes. Once restored, the applications should launch properly.
To delete applications from your iPhone, you simply press and hold on one of the icons. All the icons will then start jiggling. Simply click the "X" in the top left hand of the application and it will ask you if you want to delete it.
Note that even if you delete an application entirely from both your iPhone and your iTunes install, you can still re-download it for free. This is true no matter what the price of the application was.
