Frosty, I went to a dreadful comp in the 80s, but one of my university friends went to one of the best independents in the country. A few months ago, I actually asked her what it was like to go to a school like that.
She was really reticent about answering the question, but finally she started to talk. One of the things that really stuck out for me was that their sports options included archery and horse riding. 
Overall, what became clear was that her school focused on the application of knowledge in real life and how such knowledge fitted into the grand scheme of civilisation.
In my secondary school French lesson, we were taught to parrot sentences to pass the GCSE; my friend was taught how to speak French to a fluent standard. In art class, we were left to muck about; she was taught how to draw and sketch, and was taken on a water-colour trip to Italy.
In my English class, we read bits of a Shakespeare play. At her school, they read the whole play, a few others in the same genre, put the play on in the school theatre, and looked at the history of drama up to that point. They also put on a Greek tragedy as part of the same package of work, whilst also being taught how to write an effective essay and how to defend a hypothesis.
The upshot of this was that when I started university, it took me the first and second year just to get to the same level of knowledge that she and other privately-educated students had before they even put in a UCAS application. I had to put in a serious amount of work.
Like Jon Snow, I knew nothing. 
And something I have noticed in my own profession is the extent to which good public schools teach certain concepts, noticeably that anything is achieveable and you can therefore achieve it if you put your mind to it and find the correct process to achieve it. There is not a lot of self-doubt or self-editing, which is rife amongst my comp-educated friends to this day.
As a very rough example: publicly-educated people tend to assume that what they produce will be good or within the ball-park of "reasonable"; state-educated people tend to assume that what they produce will be shit and it will take a lot of work for it to reach the level of "okay."
Indeed, looking back at my comp education, I can now identify how limiting a lot of the language and attitudes towards the pupils in the school were -- and it still makes me cross. I went to school with a lot of very smart cookies who were let down by the poor ethos and have had to pull themselves up by their bootstraps to achieve even half of their potential.