I've never been to Davis, but I live in New York now so I can comment on that. New York is absolutely amazing - always something to do, great neighborhoods and great restaurants. At the same time, I find it to be kind of exhausting because it's crowded, dirty, and kind of aggressive. I wouldn't trade my experience of living here for anything, but I'm excited to move on in the fall when I go grad school in Minnesota or California.
As for NYU specifically, the area around NYU is one of my favorite parts of the city, but unless you're living in NYU housing (and I don't think there's much for grad students), you're probably paying $1,500 or $2,000 per month for your part of the rent with roommates. Commuting to NYU by subway from other parts of the city is really easy since most of the trains pass through the area, so you can definitely live somewhere decent and pay $800 or $900 if you have roommates, though that's still very expensive compared to most places. If you're living alone, you'll be hard pressed to find somewhere for much less than $1,100. Groceries are also more expensive and eating out and going to bars will set you back more here than most other cities. I guess all of this goes to say that New York is very expensive and it might be hard to enjoy it to its fullest while you're living on a grad school budget. If you really want to establish yourself in the Northeast, it might be worth it, but otherwise I think it's a city probably better suited to life between or after school.