There are a lot of pupils who come to my DCs state grammar school from the private sector. I am on my 3rd DC there and,for what it is worth, talking to parents over the years who have come from the independents, their main gripes are:
Much scruffier than their old school.
Holidays shorter.
Not as much discipline (but see below)
Not as much 'individual' teaching/care (ie much more sink or swim).
(Not really said out loud but a definite feeling of there are too many 'people NOT like us' there!)
As far as I am concerned, my DCs gain the following:
A good social mix of friends (from really quite wealthy parents to very hard up ones for want of a better word).
A good ethnic mix.
A good academic education.
No bullying (and hence not too much discipline at the school, isn't needed).
Mixing with children of more or less the same academic ability (not being teased for being 'clever').
I think it is very difficult to compare independent schools with state schools, other than as far as the academics are concerned. You will not get the same extras as you get in the private sector, that's what you are paying the money for...
Many of the children from the private sector find the transition to the state sector quite hard at first and find it difficult to relate to the children from very 'ordinary' backgrounds (who don't go ski-ing as a matter of course etc for want of a better example)
However, by the end of the first year, they all seem to settle down and by the end of their time there they absolutely love it and have nothing but praise for the school.