After iOS 6 no longer comes with a Youtube client, it needs its own app. Google has released a corresponding Youtube client today, but it looks like it was made with a hot needle: important functions like AirPlay are not fully implemented, there is no landscape mode until you play a video. The app was submitted before Apple released iOS 6 - while it can be assumed that Google has capable programmers, they don't seem to have had much time in advance to create an all-around usable app. Currently, the mobile Youtube website is the better alternative.
However, for those who want to use a client, "Jasmine" is recommended. This free Youtube client for iOS 6 offers everything you need and is quick and easy to use. It remains to be seen whether Google itself will follow suit and, for example, try to provide a really good Youtube app that corresponds to the Android variant.