Peter: "It actually doesn't. It must be an add-on the OP is referring to." I thought not.

And your assumption that Mozilla would not permit anything malicious is true - anything on https://addons.mozilla.org (AMO) is subject to review - the policies are on AMO.

YouTube will stream videos in two formats - Flash and 'HTML5' - and some are available in HD - so that could account for apparant differences in quality between viewed and vownloaded videos.

You can set which format you view here
https://www.youtube.com/html5