After Apple removed the YouTube app that had accompanied iOS since its early days from (the betas of) iOS 6 it was only a matter of time until Google released its own YouTube app for iOS. And today it has.
It would seem that the original removal of the app was probably mutual, with Apple not wanting to ship its devices with a Google product (especially considering that Apple's own YouTube app always kind of sucked), and Google wanting to display ads on the app's videos — something Google's own app now does.
But Google's own app is actually quite good. It's worlds better than Apple's YouTube app, and it's fast and easy to use. My one recommendation is to go into the settings and under the "YouTube" section set the "Home Feed" to show you "Uploads Only". This way you won't see any of the "liked" and "recommended" crap from YouTube and will instead see only the actually videos from the people that you actually want to watch.
The only downsides to this new app are, like mentioned above, the fact that it now shows ads (the iOS YouTube app used to be the only way to truly escape them totally), and the fact that there is no iPad version (yet — from what I hear it is in the works).