I have the same problem, but with Firefox. Embedded YouTube videos don't play in non-YouTube web pages. I can click the link in the top of the video to be taken directly to YouTube, and I can watch the video there, but when I click the big Play button in the middle of the video window, I just get a black rectangle.
I tried uninstalling Flash player as suggested above, rebooted my machine, and reinstalled Flash player, but that did not fix the problem. This is a non-Flash player problem that Apple should look into. I have not tried this on other browsers to see if I have the same problem there or not, as I do not use other browsers besides Firefox. I would rather not have to use another browser just so that I can watch embedded YouTube videos. I would prefer if Apple (or another knowledgeable user) can post the solution to the problem. Thanks!