We did a week there in January years ago. Yes it was snowing, and yes it was feckin' cold. But dress for it and you will still have a great time. As said elsewhere, you won't have the huge queues for things as in the summer months.
Also been there in early September, and it was stiflingly hot and humid, and the streets/subways just stank with it to be honest.
One thing it isn't though, is remotely a budget trip. In most places in the US, it is expensive to eat and drink out nowadays, but NYC takes it to another level. Hotels too aren't cheap, and if they are, it'll be for good reason. Avoid, and pay as much as you can afford for a decent place.