Seems like this is the generation where game recording becomes a required feature for consoles. Both the PS4 and Xbox One have the capability to save and share gameplay clips but there's one big difference: users will be able to share Xbox One gameplay clips all over the web, including YouTube.
Microsoft's new game console hooks into the tech giant's SkyDrive and that means that your XB1 snippets will show up on the cloud service's desktop or mobile clients. It'll presumably be only a couple of clicks to move clips to YouTube, Vimeo or similar sharing sites. Keep in mind that recordings from PS4 gameplay can be shared via Facebook or with PlayStation Network friends, options that are far more limited than being able to put it where you want. Sony has said that they'll eventually have YouTube support for the PS4.
It should be noted that the Xbox One's Game DVR feature requires an Xbox Live Gold subscription. SkyDrive on the Xbox 360 also requires a Gold subscription; it's safe to assume that the same will be true with the Xbox One. Of course, the true test of how all of this video sharing works happens once the Xbox One makes its big debut this Friday.