While there's no official explanation of this move on the site, Cnet's Chris Soghoian speculates that it might have something to do with YouTube's privacy policy.
While cookies on the net are quite a common affair, with every site leaving them left and right, Google has lately been under a lot of pressure for tracking sessions even if you don't actually click on the video. In practice, it meant that if you embed a YouTube video on your site, Google will get some information on that user's browser habits even the user never actually views the embedded video.
However, to complicate matters further, YouTube has recently changed their policy on cookies for embedded videos, adding an option to enable "delayed cookies," which effectively means that
"...the YouTube video player will not set any non-session cookies on the computer of a visitor (viewing the page on which the YouTube video is embedded). The YouTube video player may set non-session cookies on the visitor's computer once the visitor clicks on the YouTube video player."
Be that as it may, the latest video posted on whitehouse.gov (embedded below) doesn't leave cookies on your computer.
The fact that Google, in addition to its new privacy options for video embedding, now has a direct (and somewhat ugly, may I add) link to its privacy policy on any video hosted on a .gov domain clearly wasn't enough for the White House. In these examples, you can see how clumsily this is done, as it's really hard to click on the embed and sharing options with the privacy info link in place.