Check settings\General\Restrictions\Allowed content is at a max for all categories - alternatively try turning restrictions off and see if that makes a difference.
The only other suggestion I can make at this time is to wonder if You Tube, Your Account is similar to other applications that have a settings page that restricts access from other locations/devices. Log in to You tube from your browser and look under my account\help
Trouble logging in from other locations/devices..
Finally, you said you had logged in from your PC first, have you logged out first, before trying to log in on your pad?
Also when you restored - did you sync to itunes afterwards to commit all your settings once you were set up. IE your apple id is correctly registered, your device is set for home sharing along with your PC,
Just check here on your PC. Open iTunes Store, at right in quick links click on Account, log in, and check both your PC and device are listed by clicking on mManage Devices. The YouTube App on your pad is managed under your iTunes/Apple sign up. Check your Country under Change Country or Region is also correctly listed and matches your existing YouTube profile information.
I cannot think of one single reason why you are having the problems you are, but it could be no more thana process of elimination. Best Wishes Ct