When it comes to actually watching videos, the latest version of the YouTube app offers new ways to continue watching a video in a smaller window while you browse. Once you find a video, you can hit play to watch, then simply touch and drag the video to the lower right corner to make it continue to play in a window. A touch and swipe back upwards will return it to regular size (without going full screen), and a drag to the left closes the video. Being able to move a playing video out of the way also means you'll be able to line up your next video in advance when you don't want to go to the trouble of making a playlist.
It's important to note that the picture-in-picture style of video watching is only in portrait orientation on the iPhone. Turning your iPhone sideways automatically goes to full screen (like the previous version of the app). On the iPad, you can use picture-in-picture mode in both portrait and landscape modes.
As always, there's a share button that lets you post the video to Google+, Twitter, or Facebook, and other options like sending via e-mail or Messages. All of the YouTube video-watching features are laid out with the touch screen in mind, and even after viewing only a few videos, it's easy to see it's a big improvement over the last version.