The state v private division that seems to be a hot topic on here is a misnomer.
Look at the actual state schools a high percentage at Cambridge come from. They are nowhere near your average comp, for the most part. Brampton Manor (where they are selected and fast-tracked from Year 10, inc 6am starts). About 80 come from there. Hills Rd 6th Form College in Cambridge (where the DC of the Cambridge academics / researchers go). Similar number again. Peter Symonds. A whole list of super-selective grammars in London, Bucks, Essex and Kent. You can count on your fingers the state schools that over half of the 'state school ' students come from.
Rather than comparing 'state v private' it's only really meaningful to look at 'super-selective' v 'selective' v 'non-selective.' There are schools in both sectors in all these categories. Somewhere like St Paul's will have more in common with a super-selective grammar like Tiffin or Latymer, than a non-selective independent school in the countryside that anyone can get into as long as you pay up. Independents vary massively - some are utterly crap. The demographic who use boarding schools are different again. You can't compare Eton to a London Day School, nor can you compare your average parochial private school with 'meh' results, but a focus on other things like sports or pastoral care or religion. Totally different demographics in all cases.
FWIW DC is at Cambridge and says that the ones who need most 'hand holding' in the first year are the ones from comps and / or non-selective private schools where they have generally not encountered much competition and have been hailed as 'exceptional' most of their lives. The switch to becoming a small fish in a big pond comes as a shock to them as they are not used to feeling average. They also seem shocked that deadlines actually mean deadlines, as everyone was so impressed with them in the schools they came from, they've been able to get away with quite a lot. Eg. One girl went was in the local paper and in the radio for getting two A stars and an A. If you are told you are exceptional, it's hard when you realise you're not and no fuss is made. She has really struggled in her first year with all sorts of dramas. The ones from selective schools are used to nobody noticing them because top grades and competition have just been facts of life where they came from so they just hit the ground running. They are used to the pace and get on with it.
By the third year, it makes no difference what school they went to. They've all been in the same uni for two years by then and they are adults!