IOS – How to recover factory reset in iPhone 5 apps without backup

Yes, if you do a direct sync, your apps will be wiped out. You should transfer the apps to the new iTunes by doing this:

Files -> Transfer Purchases from iPod

This step should transfer your apps to iTunes. You can also see this Apple.SE post for a similar situation. After transfering purchases, you should then be able to sync with the new iTunes without losing your apps. The same situation holds for music.

Probably you didn't transfer your purchased apps to iTunes before you restored the phone. Under normal circumstances you would have gotten a warning message about this. Look under Applications in iTunes, check the iPhone tab and see if your apps are listed there, if not they are not backed up. If you have connected your iTunes software with your iPhone Apple account the apps should sync automatically.

Only way to get them back otherwise is to go to Appstore and download them again. Easiest way is to enter Appstore, go to the Update tab and press Purchases/Not on iPhone. (Using the same Apple-account you won't have to pay for your purchased apps again.)