camtmom, unless they've changed their policy recently, NYU will not allow anyone under the age of 17 stay in their dorms during the summer. The Tisch CAP21 summer program is a more 'controlled' atmosphere, I guess is the way I would put it. It's supervised, the kids are kept busy with all kinds of activities, seeing the city and various productions. It's a great program. They aren't allowed visitors in the dorms and there are curfews, so they are definitely not out 'running around unchaperoned'.
The earlier you get your app in the better, but not because it's rolling admission, it's just because if you happen to be missing even one little thing on the deadline date, they won't consider your appl. They only accept 16, I believe, so it's very competitive. The kids also get college credit for the program which is nice. I think they're both good programs if you feel your kid needs that kind of thing in the summer. Good luck!