I'd say brand of cheese is more important than type of cheese. Grande cheese is awesome, and it is what you'll find on most great New York slices. It really doesn't matter if it's part-skim mozzarella, whole milk mozzarella, fresh mozzarella, or provolone (although Grande's provolone is very strong and is probably best used as a 50/50 mix with mozzarella). Unfortunately it's not always easy to get your hands on Grande because it's usually only available by the case, from foodservice distributors. However, you can have it shipped in 5-lb bags from Vern's Cheese or Pennmac, or you might be lucky enough to find it in Italian specialty shops in some cities.
I'm sure there are some other great cheese makers, but I can always tell a huge difference in the quality of my own pizzas whenever I get my hands on Grande cheese. Some kinds of pizza don't need Grande, but to me it's not New York style pizza without Grande.