Hey Mar -
Ill first say that I just spent my last two years living in NYC, but I am originally from Minnesota. I suppose it depends on your prerogative, but I would much much much rather reside in Minny then I would NYC. Life is much slower here then the east coast. Our state nickname is"minnesota nice," which is pretty dead on. Money goes much further here then on either of the coasts as well.
In regards to schooling in NYC, if you were to look up education at a state wide level, MN would be in the top half, forgive me I do not know the statistics, but might even be in the top quarter of states...
There is plenty to do, but you will need a car. No other place on earth beats MN for 7evenish months of the year. 20,000 lakes, lots of hiking and bike trails, concerts, bbq's and such.
We are one of the first states, minus MA i believe, that is trying to globalize health care on a state wide level.
On to your real question; the winter weather. Is it cold from November till March? Yes Mam it is. Is there a lot of snow? Yep to that one too. But things don't shut down here like they do in NYC when there is snow, because we are adequately prepared. I would actually say that I like the winters better here then I did in NYC, because in NYC its the wind that cuts right through you...the wind isn't as bad here.
Now if you want nothing to do with the cold, then MN isn't for you....but if it your lil one your concerned about I wouldn't fret....Kids love the winter here. They can't wait for a big snow fall to go out and make a fort or a snow man.