Personally I prefer United to American though American is now trying to get more competitive while trying to merge. Now here for some pros and cons:
Infant Friendliness
This has been discussed on the forum ad nauseam so I won't repeat this but just refer you to the question asked previously on this forum: US to Asia via Europe with infants
One thing that I would add to the information provided there is that American Airline is now going through Restructuring and a Merger with US Airways, this usually entails layoffs during consolidation, so this always weighs on employees minds, and while they are likely to keep being professional "Freudian slips" are more likely to happen.
Also I don't see Alex Cross any more on the list of movies available from United.
Airport Convenience and Professionalism
I can't say anything about the differences in personnel behavior because I haven't found any. Friendliness is about the same in both(I am yet to find any), but the information is will be provided on what you need and you will be directed to where you need to go. As far as baggage handling is concerned and speed of clearing border control and customs I would say it's about the same. Personally I prefer Newark to JFK for location convenience but since you are only planning to stay the night and leave it won't matter much to you.
Quality of Surrounding Hotels
Depends entirely on your price range. There are hotels like Sheraton, Hilton and Marriott, to Courtyard, DoubleTree, and Hilton Garden, to HoJo, Days Inn and Ramada, to no-name motels like JFK Inn, around both airports, so the choice is yours. I would suggest picking it ahead of time and making reservations so you won't have to wait at the airport a long time for a pick up by a courtesy shuttle, which hotel normally provides for you. Hotels around JFK will be slightly more expensive since they are located inside the New York City limits and also will carry higher taxes on the Rooms.
HOTELS AMENDMENT
Since your price range is < $200 total you're probably going to cover most hotels with ease. Not being in the path of planes landing or taking off would be a little harder especially in JFK where depending on prevailing winds planes can land coming over the city to runway 22R. So you need to be away from the glide path which means that your choices might be limited to: Days Inn, Best Western, Holiday Inn Express, Comfort Inn, HoJo, JFK Inn, and a couple of others. Noise situation is better in Newark where hotels except for Marriott are located far enough away from the airport and away from glide paths I have stayed in a Courtyard on Frontage road and in EconoLodge on Route 1 and it's pretty quiet with reasonable prices but you can pick your as choices are plentiful.
