Depends on the school. There are great state schools and awful independent ones.
There is generally a greater wealth of opportunity - clubs, sports, service and fundraising etc etc in independents but whether that is worth the money or not depends on what you want from school. Remember that many of these opportunities will require additional payment. With a good state school and sports clubs, scouts/guides/cadets, music and drama lessons outside school, I am not sure there is much difference other than the overall cost.
Smaller class sizes are a plus. But may not be that different to the state sector at GCSE and A Level. In state, I regularly taught 15-20 at GCSE and around 5 - 7 at A level. I have taught 18 in independent at GCSE and up to 8 at A Level. You should have specialist teachers in an independent (I would expect your MFL, maths, physics and chemistry teachers to be genuine specialists) but frankly, recruitment is an increasing nightmare where ever you work. Your teachers may not be qualified in both sectors - increasingly in the state sector but in the independent I work in, all teachers are required to have QTS so that is a question for open days if it matters to you.
The difference, as I have seen it, is overall confidence, poise, self-belief. The majority of kids in independents seem to have this in abundance. It is noticeable. It is culturally OK to be clever, and there is competition to get the highest grades. This is not my experience in the state sector where wanting to learn is frequently viewed as some kind of character flaw. Children and young people who are a little different generally seem to fair better in independents.
Is it changing? Undoubtedly. The current climate is not conducive to good teaching and learning in the state sector. Class sizes are increasing. Supply is hard to come by to cover absence, particularly longer absence like maternity. Non-specialist teaching is increasingly common - including at KS4 and beyond. My subject only got 31% of its quota places filled with trainees this year so those teachers will have their pick of schools - if you can’t get your child into the best school locally and you can afford independent, I would suggest having a look at least. I wouldn’t have said that up to a couple of years ago but things have taken a turn for the worse. Behaviour isn’t great and increasingly difficult to manage but obviously that varies from school to school.
As an aside, I work in an independent in an area that is broadly very deprived. Our year 7 intake increased by 80% this year which is something considering the current economic situation. We have a waiting list for years 8 and 9 and additional waiting lists for three of our prep school classes. I can’t explain it other than to suggest those who can afford it are not taking the risk with the state schools even with excellent reputations where they might have done previously. Covid, lockdown learning (or lack of in some state schools), increasing mental health problems, antisocial behaviour etc etc are all taking their toll.