Will my car insurance cover driving my neighbors car?

No it will not. Auto insurance in New York follows the car not the person. So if you get into an accident with their car, it is their auto insurance that would respond to the damages. The accident, if a police report was done, would show up on the driver's motor vehicle record though. Same with a ticket, like speeding. That too would follow the drivers motor vehicle report, not the neighbors.

So don't assume that because you have auto insurance it follows your borrowed car. That only happens in a rental car because it is considered a "temporary substitute" vehicle. Usually rented for 5-6 or 7 days at a time and not a "I need to get to my doctors appointment" situation. That is totally different than borrowing your neighbors car. Just don't confuse the two situations.

Rental car, if you have comp & collision on your policy, would be a claim on your own insurance if you caused an accident. Your neighbors car would be on their car insurance policy.