Netflix wasn't working either. Did a search and found out if you have cookies disabled in safari Netflix will not work in IOS 7. I turned on cookies in safari's settings and Netflix works, and safari will now play YouTube clips.
YouTube still does not work in the Facebook app, or 3rd party browsers (Atomic Web).
I've always had cookies disabled in previous iOS's with no problem. It would be nice if apple would inform its customers about significant changes that affect basic operations.
With iOS 7 every video automatically opened in YouTube app until I removed the YouTube App. Now Safari plays videos but I often the message "sorry this video isn't available for this device". The YouTube App supports more video types. Safari supports H.264 but not Windows format nor Flash type videos.
I'll try reinstalling the YouTube App sometime later and hope it doesn't prevent Safari from playing H.264.
Hopefully the menu will offer playback choices instead of automatically launching the YouTube App.