My 'formative' years (10-21) were spent in NYC. NYC is a tough town to live in; SF is not only more beautiful to look at, but far more accommodating to different lifestyles. Manhattan forces you to live ITS lifestyle or spits you out; in San Francisco, you do your own thing and find whatever you need there to enhance it.
As far as playing outside surrounded by any semblance of 'nature', Central Park is it, for kids and adults alike. In the summer months it's overwhelmingly crowded, and still dangerous to go at night. Other than Central Park, there are 'paved' parks, block-long fenced-in areas with swings, slides, benches, and a few trees and patches of grass, where parents take their kids, but it's not much of an 'outdoor' experience.
In San Francisco proper, there are still many areas that feel like suburbia, where kids have yards (back and front) and neighborhood parks to play in. San Francisco's answer to Central Park is its own expansive and fascinating Golden Gate Park. Kids have a far greater exposure to 'nature' living in San Francisco than they do in Manhattan.
You'll never want to drive in Manhattan, unless you like pulling your hair out and yelling. But don't get on the subways either! Cabs, that's the only way to go in Manhattan, or any of the other burroughs. In contrast, everyone in SF does drive, although like most cities, congestion is a problem in SF-proper. SF has the BEST mass-transportation system, though. You can go anywhere in the Bay Area either by bus or by Bart quickly and fairly inexpensively. And once you get beyond the downtown area of SF, the traffic isn't nearly as bad as in NYC, other than during commute times.
Smog is far worse in NYC than SF. In Manhattan, all the carbon monoxide and other toxins from the cars, diesel buses and trucks, gets locked in because of the highrise buildings (which create a micro-environment unique to Manhattan). There is no escaping the fumes in Manhattan. The constant breezes and onshore flow along the west coast keeps SF pretty clean. What smog there is, and there is smog, lays over the East Bay area (Oakland).
I left NYC for San Francisco at the age of 21. I never regretted leaving and I would never go back to the East coast. Both of the cities offer wonderful cultural outlets, with NYC exceeding SF in the theatre and museum areas. But quantity doesn't necessarily make NYC a better choice as a place to live, not in my opinion anyway.
In my opinion, you, and your children especially, would be far better off choosing San Francisco. It is a much friendlier city, and you don't have to worry about kids developing that awful New York accent
. People are far friendlier in SF. New Yorkers are totally preoccupied with getting from point A to point B and there is NO 'small-talk' among strangers. You ignore one another, even in crowded elevators where you'll spend 10% of your life. The pace is much faster in NYC than in SF, and the competition to get things done, is greater too. There is far more "keeping up with the Joneses" in NYC than there is in SF.
The climate alone should sell you on SF. Do you really want to live in a city where every summer you'll experience 95-100 degree weather with 90% humidity? Let me tell, that is true hell! And in The City itself, it feels even hotter, because of all the highrise office buildings and that micro-climate I mentioned above. The heat gets locked in and it's hotter in Manhattan than elsewhere in NYC. It's horrible, believe me! Of course, most people who live in NYC attempt to leave at least for the month of August, if they can afford it.
The winters can be really rough too, if you don't like cold and snow and all. The winters never bothered me as much but it can be pretty brutal. SF can be very chilly in the summer, though (recall Mark Twain's comment? "The coldest winter I ever spent was a summer in San Francisco.") and you don't have the 'beach-going' opportunities that New York provides (but NOT Manhattan! Manhattan is an island, but you seldom see the water while your there, only from a distance in highrises, or while travelling some of the arteries. The most accessible water front in Manhattan is Seaport which is in the southern part of Manhattan, beyond Wall Street.). If you want to go to the beach in Manhattan, be willing to drive two to four hours to get there.
There is far more access to ocean and water front viewing in San Francisco than in Manhattan. It's a breeze to get to the Marina area of SF, and walk along the shore front, overlooking the Golden Gate Bridge and Alcatraz Island. Or hop on the trolley to get to the South of Market area with all its various ports. Travel west along Geary Street to the Cliff House and the Pacific Ocean, which you can walk for miles along beautiful clean sandy beaches that are never crowded. Explore the fascinating ruins of the old Sutro baths while there. Swimming, however, is only for those sporting wet suits or in the very hottest times of late summer, when the water may warm up enough to enjoy a quick dip.
In Manhattan, getting anywhere away from The City takes you two-three hours, at least. In San Francisco, you drive 20 minutes over the Golden Gate Bridge and you're in the magnificent Golden Gate Recreational area with its fabulous hiking trails and secluded beaches. If you like winter sports, you can travel another hour (three total) to the Sierra Nevada mountains for fabulous skiing at numerous of resorts. In Manhattan, you travel about the same amount of time (2-1/2 to 3 hours) to get to Hunters Mountain... or Hunter's BUMP! East coast skiing cannot compare on any level to west coast.
If you're used to the Pacific Northwest environment, living in San Francisco will be far more comfortable for you. You'll have a huge adjustment moving to New York City, and frankly, I doubt you'll like it.