The Christmas season is one of the heaviest visited for NYC, so "bargains" as far as hotels, will be hard to find. My advice is to book ASAP, as anywhere near that price range (think about $150.+ for something that isn't a dump) will book fast. You may consider that less expensive places are going to be an hour subway ride or more from the outer boroughs.
I prefer JFK, as it is directly on a subway line, but that's just me.
Snow... hmm. It's a roll of the dice. In recent years, except for freak storms in the Fall and Jan/Feb, very few have yielded snow. NYC has plenty to see, and do, and is a lot less filthy without snow. Snow storms make a real mess, and grumpy people.