New iPhone, took a few nice pictures, then restored with the data of old iPhone. Pictures gone - help!
When you restored the phone, you have changed the encryption keys on the iPhone and the old keys are gone. Unfortunately, the chances of recovering those pictures are almost 0%. It is not simply not worth trying.

The apps are there because when you've downloaded them, you have been logged in to the App Store with your account, so the information about your download/purchase was stored on Apple's server. After you have restored your phone, you have logged in again with the same account, and thus it is possible for the phone to restore your purchases/downloads.