Youtube videos are typically rubbish in quality.
I've noticed issues with Youtube videos, but it normally has to do with the Youtube player just not responding. Usually at least for me it goes to whatever "HD" resolution it supports when I go full screen or clears up as it continues.
Granted it could all be 480P.
Download McTube it lets you set up to 1080p for Full HD videos while on cellular data... but for embedded you are out of luck! Some embedded videos don't offer any quality option, but others do... but not beyond 480p, so you have to link up to McTube if you want full quality... most annoying!
Really?
I find them excellent quality, but of course they are dependent on the quality and the device the poster uploaded them at, but 1080p youtube's are generally lovely and clear.