I have lived in Edinburgh or just outside for 50 years. It can get cold, but more in a damp raw way rather than cold or crispy.
It is a much darker place than New York, ( 55 degrees North rather than NY 40 degrees) winter days are much longer.
Edinburgh is a very vibrant cosmopolitan city- almost no racial tension, lots of tourists, and lots of great arts, restaurants and activities. Big academic life middle class eco types- which makes for great activities and acceptance of children. The city is compact, you can easily drive the width of it in half an hour. Unspoilt countryside is all around- hills, beaches, big artificial ski slope. Trams take you from the city to the airport- althouigh it is only a few miles.
Living in the city is expensive. Morningside/stockbridge/Bruntsfield are all great for families but costly.
Move just a few miles outside the city and prices drop dramatically. Public transport is good from all directions into the city- the East coast is well served by a good rail link having you to unspoiled costal towns within minutes of the city centre- but prices along this rail link are high.
A good buy may be somewhere on the new Galashiels to Edinburgh rail link which is being opened in 2015, with many planned stations just a few minutes to Edinburgh.
Property prices along this link are still very low you could snap up a bargain as when the rail link opens prices will soar.
I have just bought a 5 bedroomed house for £200K with a garden close to ameneties in a quiet leafy suberb which is a 2 minute walk to one of the new stations. The proposed journey time is 14 minutes from here to Waverley station in Princes Street, right in the heart of the city.
I love it here but then I am biased!