There is no magic answer on dealing with that noisy inconsiderate (and probably contemptuous) neighbor or with noise coming from outside your building, but there are strategies you can develop.
Many tenants think there's a law that allows a neighbor to be noisy up till 10 or 11 at night. There's not. But common sense and practical application of the NYC Noise Code recognizes that a reasonable level of noise is acceptable during daytime hours. On the other hand, a rock band reverberating in your apartment, even at 12 noon, is probably unacceptable and you have a right to complain to the police, to your landlord and to the NYC Department of Environmental Protection. Which is the best is hard to say since every situation is different. And of course, talking with your neighbor should be the first step.
You must be prepared to push the issue. If the police don't respond, call back and get the name of the desk officer and complain to his/her Captain [and higher]. Call the Community Board and your local politicians. But while you're being persistant, don't come off as a crazy. And after you've done your homework and documented the problem, complain to your landlord and be prepared to withhold your rent to force him to bring a nuisance action against the offending neighbor. If you decide to take this route, make sure you understand how to deal with Housing Court.
The following articles discuss noise issues and possible strategies on dealing with the problem. Be aware that some of the agency addresses and phone numbers might be out-of-date.