bit late coming to this thread but these are my observations. Sorry it's a bit long:
it is kind of trite to say but it all depends on what you value most out of education/life and what kind of child you have.
If you value academic success above all then in the vast majority of cases your child is likely to get better exam results at a private school as opposed to a state school. (unless you're "lucky" enough to get into a grammar school)
If your child is bright then (IME) they will do fine pretty much anywhere as long as they are happy, and you may end up paying a pretty substantial premium for marginally improved exam results. If your child is not that academically bright then tbh a competitive private school might be hell on earth.
If your child is average academically then you are likely to get the most improvement in grades for your money.
(BTW you should bear in mind that employers/universities will likely take into account the school when they assess just how smart your child is from their exam grades so it may all be canceleed out..)
In terms of confidence - again a confident child will probably do all right anywhere, whereas a shy or diffident child may be overwhelmed by a rufty tufty state school. BUT bullying can happen anywhere and there is just as much pressure to fit in at private schools. There is obviously a wider social mix at state schools and (IMO) better preparation for the real world...
Also important I think is your attitude to competitiveness. If you think competitiveness is a Good Thing and splendid preparation for life then IME private schools are more academically competitive but also more socially competitive. I know some children whose parents scraped together the money to send them to private school who had the most miserable time imaginable because their clothes/holidays/cars/houses were Not Up To Scratch.
No one seems to have mentioned either that private schools are almost always single sex. You might think this is a good thing but (IMO and IME) it is a bizarre and unnatural idea to separate out the sexes and doesn't lead to healthy relationships and attitudes.
School is just one part of the mix and what happens at home is every bit as important. For me the most important thing would be that your child is enjoying school and isn't wanting to leg it out of the school gate at the first opportunity.