Whew, since Manhattan brought it up first, I can go ahead and make a case for Brooklyn without fearing for my life. Since it seems like Toddlernym is at the age where you are more interested in a backyard than in the school system, Brooklyn might be an ideal choice for you. If we can get Ike in this thread, I think he could tell you all the joys of living in Park Slope (Brooklyn). If Mr. Nym is working in say, the Wall Street area, then the commute from Brooklyn would be about the same as the commute from the Upper West Side.
If you are planning to move avec your furniture, then a Park Slope brownstone would actually hold all your furniture, probably better than an apartment in the same price range in Manhattan.
Plus, it is a bit easier to have a car in Brooklyn, should you want one.
In a more abstract sense, Brooklyn is a joy. It is so close to Manhattan, yet it has its own flavor and unique neighborhoods. A few years ago, lists of “hot bars” in New York Magazine would have been exclusively in Manhattan, now bars, cafes, restaurants and shops in Brooklyn are getting extensive city-wide coverage. Brooklyn has Prospect Park, the Brooklyn Museum of Art (see great art AND piss off the Mayor, ya can’t beat that with a stick), BAM, and the Botanic Garden. And they’re doing fantastic things with the Esplanade.
I lived in Manhattan for ten years, and swore I would never move out. Then I moved to Brooklyn, and cried because I hadn’t moved ten years earlier.
Is the company getting you a realtor? That’s the best way to find apartments, especially since you will have limited time to make a decision. Insist on a good realtor. Corcoran is good (sort of, but if you’re not actually the ones paying them, it doesn’t matter) and will be able to show you things in both Brooklyn and Manhattan.