It is a complicated topic not helped by the data being completely reversed by mistake leading to the headlines that state did better than private when it was vice versa for that study as data had been reversed - see Mia85 post above.
Other complications are some areas of the country have very academic selective and grammar schools whereas areas like Newcastle where I am from went all comprehensive since 1970. Also 7% of children go private but 20% at sixth form level (and I think half of UK children don't even get 5 decent GCSEs and plenty are out of school by 16 into some kind of apprenticeship etc etc)
The bottom line is for me I was happy to pay school fees for a vast number of reasons and plenty of parents where I live choose good London state schools instead and it can be a choice not just a money issue.
The differentiator tends to be as someone said above which children (of those bright enough for university ) are self starters (and which are lazy as sin and you get the lazy lot in both state and private schools).
It is very hard to study this area as there are a raft of factors from tutoring for those in the state sector, state grammars, rich state school parents v very poor private school sixth form pupils on full scholarships (not very many of the latter of course...)
Most employers in the higher paid graduate jobs try to ensure they are fair, most have some kind of contextual programme and some although having minimum A level requirements and a degree will put candidates before the interview panel or assessment day who will not know where those people are from - Oxbridge, Sunderland University, private school or not.
However we should try to track the data as it is quite interesting where people end up . My own profession, law, has compulsory annual requirements to give data about things like race and background and if you want to private school. Some bodies like I think accountancy firms will look at what kind of job did the main wage earner of your home do when you were 14 (in my case that answer is my father was a doctor, NHS consultant)