the app needs to be updated, even it the OS supports a new feature, the individual app developers still need to code those features into their apps.
especially with the short time between when apple released the "gold master" (developer pre-release of final version) and when the final version was released to public. Typically they get a week or two, this time they got less than 24 hours.
OS changes can break apps in silly and unexpected ways, YouTube (and all other developers) need time to fully bug test their apps against the actual OS people will be using. otherwise their apps might have problems that didn't exist in the previous beta version of the OS.
And if they released buggy software, the outrage would be even higher than a late release software.
Apple also tends to add a few features in the final version that weren't in the betas that they wanted to keep them secret, like the new watch faces, which can alter the way things work. (watch faces should cause much of a problem, just an easy to see example)
I'm also not sure when apple starts accepting apps for the store that have been coded for the new OS.
and most probably wouldn't submit till gold master comes out, so there's probably a back log of apps waiting for approval. Although google probably has a bit more sway and can jump the line a bit.