I downloaded "Jasmine" and it's WAY better than the youtube app for iOS. Full screen vids, will continue playing music if you exit the app and start doing something else, and just a much better experience overall.
I like the YouTube app interface, but I noticed that you can't do something that I would do frequently through YouTube on Safari which was that I would play a video, lock the phone while the video was playing (which would stop the video), and I could double tap the home button and press the play button to resume the audio. This would allow me to listen to a song on YouTube while my phone was locked.
You can't do this anymore through the YouTube app though, and you also can't play videos in full screen on the iPhone 5 without the black borders which sucks.