I will only talk about the one I know, which is the upper west side. I loved it!
People DO work very very long hours here and I suggest thinking hard about any commute over 45 minutes - does that mean Daddy will never ever get home for bedtime, etc? Is that worth it?
Also, you and your kids may love the city experience - there's an energy to Manhattan (and a convenience, with the subway and buses and lots of things to do) which I've never known in any other place. It was much easier to get DS in and out and around in Manhattan without a car than I expected - really not that hard, and masses of things within walking distance.
If you live in UWS you may well get a building with a pool. So not in your garden exactly, but just up/down in a lift, and also may have some built-in-friends in there already... worth considering. Also, because of the extremely cold winters (-10 during the day sometimes for all of Jan and Feb) and extremely hot summers (and HUMID oh my goodness especially June and July) you actually wouldn't be out in the garden much for 4-5 months of the year. I've got one now and I'm not in it from 10am-4pm because it's too hot for my toddler (even with shade) - I end up driving to indoor play areas or garden centers with aircon, so he can run around! So an indoor gym, children's playroom, or pool that's part of your apartment block is worth considering if you have little ones who get cabin fever. Buildings-with-amenities are really one of the perks of flats, if you've got the $$$$ to live in one.
And you have Riverside park on your doorstep, which isn't a back garden but IS one of the loveliest parks around. Lots of play areas (not sure how old your kids are?) including splash sites and sprinklers in the summer. The UWS is very like park slope in that it's insanely family-full, absolutely chocablock with kids, things for kids, and kid-friendly parks, hairdressers, restaurants, whathaveyou.
I used to live there. Couldn't afford it once I had kids (hah!) but it is a gorgeous place. Less trendy than Park Slope/ Brooklyn heights, but excellent schools nearby. As company is paying for private, you're going to have a LOT of choice. (Wherever you go) which is nice.