Just to give you both flip-sides... I went to private primary and secondary school. The primary was actually worse for snobbery than the secondary, but mainly because it was so very small (at one point there were ten people in my entire year group) and poor opinions of one particular girls parents (basically because they were considered common) filtered down to the kids, and she was never invited around for play dates and was sometimes teased at school. When my dad found out he was furious and was invited around our house all the time!!
At my secondary school (all girls) there were definately girls that were very rich (their own branded clothes!! and some of princess diana's god children), and there were a lot more people whose parents just got by and were trying to avoid bad state schools. It was actually very rare that the truly rich kids would bully anyone, and no one would ever say anything to anybody who said oh - I can't go on that trip it's too expensive.
Good news, in private schools, most extra curricular activities and days out are free! It's only if it's not on a school day usually that you would have to pay, and then there is usually only one 'big' trip a year, and nobody goes on all of them! At my school they ranged in price from less than a hundred (two or three day breaks to europe) a few hundred pounds (a week long european adventure holiday at a beach or skiing) to more than a thousand pounds (to go to the amazon!)
It's a bit hard to explain but it will probably be noticed if your kids don't have expensive mobile phones or iPods, and noticed again if they ask for expensive things for birthdays/christmas and don't get it. But at the same time it will only actually be a problem if your DC desperately wanted to be in with the very-small-crowd-of-people who actually care about these things. This crowd of people probably aren't actually the popular people but the ones who would like to be IYSWIM!! Good social qualities are far more likely to be valued by their peers than wealth.
Like somebody else said though, it depends which school you sent them too. Definately check it out first, preferably on a school day, if you can listen to the way the kids talk to each other in the hallways etc or ask the staff about how much of a social mix there is/what provision there is for needs based scholarships which should give you an idea of how mixed it is. Be aware that some schools, particularly those that consider themselves 'public' schools can be very very snooty. At secondary school we were shocked to go on a school trip with another private school and after years of being teased by local state school kids for being posh, to be called common by them! Apparently the teachers were equally weird/snobbish, because our teachers got really angry with their attitude to everything (ie the idea that they wouldn't have a word with their students for being so twattish as to call anybody common) and never arranged another event with them again!