the difference between Firefox and Google Chrome is that Firefox by default uses the HTML5 technology to stream youtube videos (at least it does in my version, I think: Lightweight Edition released a couple of days ago. But I'm not sure because even if I turn HTML5 off in Firefox, I'm still able to play most of the videos
) while Google Chrome has a flash plugin integrated by default in the browser in order to play them. So the source of your problem might come from the flash plugin.
Try installing the chromium browser and the separate flash plugin package and see if that makes a difference. Chromium is basically the same as Google Chrome.
apt-get install chromium-browser chromium-codecs-ffmpeg chromium-codecs-ffmpeg-extra flashplugin-installer
BTW my flash player version in Google Chrome is 22.214.171.124 and everything works fine. What is yours, please?