Another problem, another solution.
Youtube videos REFUSED to play in Safari 6 on my Mac, but played fine in Chrome. Did some digging on the 'net, and found a forum support note that said my "Click to Plugin" extension for Safari might need an update to version 2.7+ (I was on version 2.6)
I downloaded/installed the update from Macupdate:
...and now Youtube videos in Safari play just fine.
If you don't know what "Click to Plugin is, and wonder why I have it installed in the first place, you can read about it on the Macupdate link, above, or just google it! :)