I just had a rather frustrating experience with my iPhone restoring from a previous backup. Sometimes instructions are lost in translation when describing a user action—and that's what happened to me.
When you plug your iPhone into your computer, there's 2 buttons that stand out. "Check for Update" is very obvious, but the little "Restore" button below appears to be a maintenance aid when solving issues you might be having.
If you are experiencing problems with your iPhone, you can restore its original settings by clicking Restore.
I have done this many times over the years with various iPhone's and iPad's. The simple theory is this: backup your device with iTunes first, transfer any purchases you need, and you have a file in your system to revert back to. After you "Restore" to the factory settings, this is the last backup that iTunes will then apply to your "fresh" device so everything is as it was.
This is where the problem comes in! (or the appearance of one)
After you successfully backup, restore and then restore from backup, you think everything's fine. Then your device shows in iTunes with the same message: "This computer has previously been synced with an iPhone or iPod" and asks you to "Set up as a new iPhone" or "Restore from the backup of:" and the same drop down list you picked before. Didn't I just restore from my last backup? Did something go wrong?
Upon inspection, your phone seems to have all photos, email accounts, texts, etc copied back—but no Apps! The job appears to have been completed halfway!
And the revelation
After your iPhone has come back with the partial restore, iTunes isn't being as forth coming as it should. The option for "Set up as new iPhone" seems scary, since there is no reassuring description of its action or a help to read more. You are given the ability to name your device again before it finishes the job of the restore. That's it! I renamed mine so I could manually go and delete some old backups (with the old name) from the iTunes preferences. This is really just the second step to restore and all of your apps, videos, etc will continue to copy back to your phone.
"Set up as a new iPhone" should be translated to "Name your device again or keep the same dang thing."