I have children in the state and private school systems.
The things I think the private school gives my child over the things the state school gives his twin are;
An expectation that he question and challenge things. Not just academically, but also how things work, what doesn't work, why, how could it be better, how can you make it better, with who, how will you start .... a feeling that he as much a leader in his school as anyone else, not just part of the system.
A sense of pride and self-worth. Smart, high quality uniform. Historic buildings. Structured relationship with the Church.
Responsibility for himself and his time and his belongings. Expected to be in the right place at the right time doing the right thing and with all the right equipment, without much instruction and no mollycoddling. If he gets it wrong then they support him to figure out why, find his own solution for that day, make a plan so it doesn't happen again, and put in a review date to discuss whether the plan is helping.
They don't turn a blind eye. If there's a rule, then they notice if it's not kept to. The state school says no leggings, but at the end of the day they don't have the time or resources to tackle the fact that half the girls don't own anything except leggings. IMO they do the right thing by not punishing those girls, but ultimately that just means the rule is meaningless. In private school they notice if you don't have the right things; first time you're sent to the shop to get a new one and they add it to the parents bill; second time kicks off the "solve you're own problems" approach to making sure it doesn't happen again.
The pace of life is different. Private school is fast paced, 6 days per week, plus homework and sport in addition. They get used to doing a lot of things all the time. State school is 9-3 Monday to Friday and a lot of free time.
Just some thoughts ...