The Sutton Trust reports (PDF) that state school pupils do not have the same chance of getting into top universities even if they have the same grades as a student from a private school. It is important to understand that this doesn't necessarily mean that the universities are discriminating.
There is plenty of evidence that state schools are teaching to the test. That instead of equipping students with the broad range of skills they need to progress in further education and work they are focussing too much on exam results that boost the league tables. This means that, with a given quality of exam results, a privately educated pupil - where parental pressure provides an incentive for schools to provide a broader education - will have a more diverse set of skills and will usually be better prepared for university.