The player recently tested by YouTube is now live for everyone. YouTube's redesigned player has a lot in common with the new YouTube interface: it's simpler and more subtle. Controls fade out if you don't move your mouse, the progress bar is less visible, volume controls are now horizontal. It takes some time to get used to it, but the new player is one of the most brave attempts to minimize complexity in YouTube's user interface.
Unfortunately, the new video player has too many moving parts and the animation effects could become annoying. Vimeo's player is even more streamlined and more user-friendly.
Update: The new player is not available for all videos. Here's an example of video that uses the new player.
Update 2: YouTube says that it has "released a new player design for all videos without ads. The goal is for the player to be as subtle as possible so that the video itself shines and doesn't have to compete with the stuff around it. We'll be following this roll-out with player upgrades for videos with ads and then for embedded videos."