Youtube likes are classed as "social" interaction (which is good at the moment). Since the Panda / Penguin updates, social signals have gained weight as they are (allegedly!) deemed to be more representative than backlinks and less open to manipulation LOL.
If your Youtube video is really popular and getting likes (lots of "social signals"), then the value of it's link to your site has to be of higher value than one from a video with hardly any hits and no social actions.
As for the relevancy bit, I cannot give you a definitive answer.