Well, I started WW3 by asking a similar question last year, so i thought i'd chip in. DD goes to the nursery of a private day school. I was privately educated from 2+ and went to a range of schools from the £20k per year establishments that everyone's heard of, to smaller local day schools. I swore blind that I'd send my kids to state schools until we looked round the local ones. We live in the NW btw.
We were concerned by the poor levels of literacy amongst the staff, prospectuses were full of spelling and grammar errors, so were the displays and so on in the school. We were also concerned that staff seemed to think that if a child reached the "adequate" SAT level, then their job was done.
I'm embarrassed to say we looked at about 14 schools before we found DD's school. It is private, but attached to a state grammar school. The 4x4 parents are ina minority, in DD's class there are several university workers (including DH and me), plasterers, mortgage advisors, all very "average". We live in a very "white" area, so her school has one of the highest rates of ethnic minority pupils in the borough. Anyway, when we looked round it just felt "right", DD liked it so much she stayed in the nursery whilst we did the grand tour.
Sorry, none of this really answers your question, but I think it's more to do with finding then right school for you and your child than either sector having particular advantages and disadvantages.