Private schools do get spend a lot more money per student than state schools. That has to have effect on final grades. I'm guessing they really push you, if you fall behind at a private school.
I'm currently doing a teacher-training placement at a very expensive private school (one of the most expensive ones in this country).
The progress of every pupil is kept in check, reports happen every half term, a 'B' is not considered 'good' (let's face it, their parents are paying >£9k a term, what's the point if their kids don't get A/A*s?), extra-curricular activity is pushed /hard/ as well as tons of homework.
The school day runs from 0820 to 1640, with clubs running until 6pm. They have allocated homework ('prep') time every evening. There is school on Saturday mornings.
Most of the pupils (if not all) have aspirations to university - the 'good' ones - and to becoming doctors, lawyers, etc. If they don't have these aspirations themselves - their parents sure do.
20 pupils per class is considered a lot. Most vary between 15 and 18 - with classes getting smaller (sometimes much smaller) in 6th form - which gives you time to speak to pupils individually within a lesson - impossible when you have 30+ kids in one room for an hour.
So, they do squeeze each pupils of everything they can give BUT... that doesn't mean state schools don't also do a lot of the above.
AND, many have burned out by 6th form. I've seen the ones that have had enough - they're tired, their eyes are glazed-over, concentration and application to their subjects drops off.
You'd be amazed at how often a teacher or two will have had a conversation about you specifically - whatever kind of school you're at - worrying about your progress, comparing notes, trying to work out how best to motivate you. They care, these people - they wouldn't do it if they didn't.