Which school is better to go for my Bachelors in Nursing in NYC?

It is not at all worthwhile to pay the high tuition fee at NYU.

If you live in the Bronx go to Lehman. Your degree will say BSN just like NYU and other nursing school's in the US.

If you excel there, you may wish to consider going to a research university for your MSN but that is down the road several years.

One major consideration - NYU is really expensive. RNs make decent money but not if you have a large portion of your pay going to huge school loans. Keep your costs down as much as possible.

NYU is "better" in that is one of the top nursing schools in the country. Admissions will be very competitive, however, and tuition is astronomical. Lehman tuition is probably 1/10th the price or less.

Also find out whether Lehman has a two-step admissions process. At NYU, you apply to the nursing school, and once you're in, you're in; but many community colleges have a two-step program, in which you need to take general and prerequisite classes at the college first, then competitively apply  to the nursing program before you can take any nursing classes (I'm not sure Lehman is like that but many CCs are).

Anyway, don't borrow trouble. Apply to both, and if you get into both, that's the time to worry. That's also the time to visit both schools to see if you like them, and to talk to officials. "After the tour you just gave me I'm just in love with NYU, but I have to seriously consider Lehman because the tuition is so much more affordable . . . " might prod them into a tuition cut if they like you (that's how I got a top-20 law school education at a state-school rate).

Bachelors in Nursing in NYC is expensive and competitive too. If you are able to get in there you are welcome.