Not all private schools are necessarily that great. Just like state schools I am assuming there will be some private schools that are a bit crap. I don't think it makes you a snob, or necessarily smarter or anything. It does show you are wealthy and of course people get jealous. They feel like you are incredibly lucky and wish they had had the same opportunity.
I went to a state grammar school, and it's extremely good (or at least it used to be) and I'm sure it's as good as many private schools. So it really does just depend on the individual schools. If the state schools were good in my area, when I have children I'd send them there and save the money. If I was rich and the private schools were better then I'd try and send them there.