Google bundles their own build of Flash in with Chrome, the normal Flash plugin doesn't work with Chrome. It's the only browser on Linux that still gets new (non-security updates) versions of Flash.
I wish this was working properly. I am using the HTML5 trial and embedded videos will randomly not work, and many, many other videos will simply show static with "You need Adobe Flash to view this video."
For some strange reason, the exact same videos work fine on my iOS devices — just not on a no-flash desktop web browser. I'm suspicious that YouTube purposely allows a larger selection of their content to work with iOS using HTML5, and purposely fails/requires flash on a desktop under HTML5.
I get the opposite, I get videos that play fine for a web-browser, but won't work at all on mobile, even though it should. I've even taken a video, uploaded to my account and it STILL says "the owner has indicated that this video not show on mobile" or words to that effect. Though when I uploaded the video, there was no setting to click to make it unavailable to mobile users.
Besides, they've been moving YouTube to HTML5.