These are my thoughts based on experience of both (not in order of importance!)
1] Private school seems to communicate better: they release the calendar for the whole year. State school told about events a week in advance.
2] More opportunities and encouragement at Private for the quiet, non-volunteering child to be involved in things and therefore become more confident.
3] State school is restricted by the National Curriculum as to what children can learn. A bright child isn’t allow to progress outside the curriculum for their year group (although supposedly this is countered by learning more in depth about that level). Private seems to be more open to learning at the pace of the children without these constraints (but obviously they are still taught to the ability of the class to some extent).
4] Comparison- private school seems to be much more open to telling children how they are doing in comparison to the rest of the class. This could be a good or bad thing depending on the child. If they are middle ranking and competative It is a good thing.
5] Opportunities- private gives more options for participating in extra curricular activities. ie. all children given the opportunity to do DofE award, as opposed to only the first 50 to apply.
6] less expensive trips than state. This is the weird one! The state school has trips to Harry Potter World, Disneyland Paris, Iceland, Skiing etc. The private seems to just go to the local museum. Bizarrely the state also seems to be more into tech (iPads, VR headsets, interactive whiteboards) than the Private school.
7] money. Despite what everyone says on MN, children at private discuss wealth. They might not bully the poorer children, but they know who they are.
But basically as everyone has said it’s all to do with the specific child and the specific school options. Private isn’t better than State. But some Private schools can benefit some children. It’s the fit between child and school that’s important, not the money.