The look and feel of the app has changed quite a bit, but probably the most notable change is that new YouTube app bring a more full-featured guide that pops out from the left side of the screen. Within the guide, users can access recent updates from their subscriptions and recommendations on what to watch, along with the search and history functions.
Additionally, the new YouTube app adds a search bar to the top of the page, and search results will now include channels in addition to relavant videos. Speaking of channels, it looks like channel pages have gotten a bit of a refresh as well, providing a list of playlists and videos put together by the channel owners.
If you'd like to snag the revamped YouTube app for your Xbox 360, you should be able to snag it from the Xbox app store now.
Source: YouTube (Google+)