Pizza sauce (NY style)- uncooked, unprocessed, fresh, bright, light (fat free) with very little herbs.
Spaghetti sauce- slow cooked, sweet, rich, layered, hearty
Good puree is bright light red, paste is dark deep red. Puree (and peeled tomatoes) makes good pizza sauce and paste makes good spaghetti sauce.
Generally speaking. I've had pasta with an uncooked bright tasting sauce that was great. With most Italian, old fashioned families, though, gravy is simmered- for a while.
Sauce is very personal/very subjective. I feel pretty strongly that NY style sauce should be uncooked/bright and relatively simple, but, beyond that, there's as many different sauces as there are pizzerias. As I mentioned before, I try to take a fairly historic approach. I believe that when pizza crossed the water, the basil that used to go on top of the pie started going into the sauce. That's one core ingredient. Fresh basil-dried basil is an abomination. I also believe that as immigrants started using less sweet tomatoes, sugar entered the picture. Sugar is definitely another core ingredient (for non San Marzano tomatoes/puree). Salt's a no brainer. Oregano... yes, oregano is a player, although all of my favorite places go with an extremely light hand. Oregano, imo, is very easily abused. Summing up:
all define NY style pizza. Garlic is a tough call. My favorite spot adds raw garlic, so I follow their lead in my own sauce, but I haven't recognized garlic in enough NJ/NY pizzerias to be able to say "garlic defines NY pizza." I wouldn't make a pizza sauce without raw garlic, and, if I ate at a pizzeria that didn't add it, I'd probably say it was missing something. That's just me, though.
As far as potential ingredients for a classic NY style sauce- that's the list. There's many paths to pizza nirvana, though, and people make great sauces from just about anything, but, as far as my experience growing up eating NY pizza, this is it.