I do use NoScript, and Firebug, which provide additional insight into the issue. Apparently there was a recent design change, and the Flash player now is inserted by a script hosted on a server in the ytimg.com domain rather than the youtube.com domain. Once I allow scripts for ytimg.com in NoScript and reload the page, the movie starts.
If you don't use NoScript, your security software may require that you approve this domain, or you might have it blocked in your hosts file. If the domain is primarily used for advertising, then an ad-blocking extension might be blocking it.
Does any of that help?