I have close experience of both so I also have strong opinions
This is how I see it:
1) The academic education in most modern private schools is significantly better than state. Class sizes are smaller. Teaching resources higher. To be even clearer, bright pupils are stretched and the middle ground all tend to do better than they would in a state. The league tables bear this out.
You can also see this with the percentage of kids from the excellent local states who then fail to get into the private secondaries. I am not saying those children are less able, just that when the test comes around the private kids know more.
2) For some children the sporting activities at private schools are worth their weight in gold. If you have a sporty boy or girl (but mainly boy) they will play games every day with swimming clubs, cricket clubs, rugby clubs, tours and the like and that is in PREP school.
However, and this is the big kicker, I am not convinced that all of this coaching and conditioning actually matters in the long run.
I am a big fan of private schools but for A Levels, when the drive and natural intelligence of the child is the over-riding factor, I am not sure that it makes a big difference.
There is one caveat to this, however.
If a child is in an environment where it is not cool to learn (big city state school?) then they will end up being REALLY damaged by a state education. If they shy away from reading and it crushes their natural curiosity, then the child is in deep deep trouble.