We moved from London to NYC 4 years ago, we live in Battery Park City which we love. It's very family-friendly, quieter than most of the city, great parks and playgrounds. When you're thinking about where to live, how do you see yourself living? Have you spent much time in NY before? Pretty much all of Manhattan is very urban - it is a full on city, much more so than London. In Manhattan you are very unlikely to have a house with a garden, unless you are squillionaires. But a lot of apartment buildings have great facilities (children's playrooms, swimming pools and gyms, bowling alleys...) The range of activities available is huge, it's a very rich cultural experience, we have loved it. And I always feel safe in Manhattan.
Greenwich CT is nothing like that, it's a very nice town, maybe compares with Cheltenham. And your dh would have a train journey but it shouldn't be too bad, especially if he's working in midtown - he would come in to Grand Central Station which is at 42nd St.
Between the two you have some lovely neighbourhoods in Brooklyn, which are I'd say more comparable with the zone 3-type areas of London - lots of terraced houses with smallish gardens, nice communities.
Private schools in the city tend to be competitive, and the official admissions process for September has already finished. That doesn't mean you won't be able to find a place, but I'd suggest trying to start this process as quickly as possible. If your dh's relocation package includes a schools consultant they will help you research options. This is my dcs' school, we have been very very happy there.
Good luck! It's really overwhelming planning a move but it will all work out fine for you. I'm really happy with the choices we made. But I'm also quite sure that we could have made radically different choices and still ended up just as contented.