the winters are a bit colder (not much), and the summers are definitely not as hot as houston; for every day it gets to 95 in birmingham, it probably gets to 100 in houston. doesn't sound like a big deal... but those triple digits down in houston start to take their toll i bet
the best public schools in the birmingham area are vestavia hills and mountain brook. these areas are pretty expensive to live, but if you can afford it... they are great communities. tons of activities for kids like sports and whatnot... very family oriented communities. they are also the closest suburbs to the city of birmingham, so commutes are not bad.
also good are homewood and hoover. homewood is pretty expensive and very convenient to downtown, while hoover is bigger, a little further away from downtown, cheaper, and a little more diverse than vestavia/mountain brook
there are a handful of private schools worth looking at, though not as many as you might think.
john carroll catholic is a catholic high school (obviously), though not everyone who goes there is catholic (go figure).
briarwood is another christian high school in the area, though i personally don't think it's all it's cracked up to be. personally, i think the public high schools in the area (hoover, oak mountain, vestavia hills, etc.) are better than briarwood in terms of educational quality.
if possible, rent in one of these communities for 3-6 months if you can while you learn the area before you take the plunge and buy. i know if i was moving to a new city, i would definitely want to learn the lay of the land and the communities.