My experience from both state and private with my DC.
Private expected a lot of academic learning was expected to be done and solidified outside the school (more homework, tutoring from reception, etc was rampant). They didn't really teach but checked the knowledge.
The above freed up time to do a lot of other "enriching activities" - properly rehearsed and costumed school plays, multiple school trips, age-appropriate talks at school starting from, again, reception.
Theatrics around raising caps and shaking hands, straight backs, looking into eyes, etc. Children were very confident and not afraid of talking to teachers "as equals".
And then, the huge factor were families who had all additional holidays and means to take children abroad, pay for all extracurriculars ,and so on.
State school - the teachers were actually teaching the "hard and boring stuff" and not outsourcing that to parents and tutors.
The distance between teachers, headteachers and children was huge, and seemed a bit condescending.
Commenting on children in the presence of children was a norm (in our PS they would never criticise the child in front of parents).
Miraculous how much the school could do and organise given the funding constraint. I am truly impressed.
Parent involvement was definitely less, e.g. few very involved and active parents, many were MIA, silent class WhatsApp and so on.
As a bottom line, as a pure academic investment I don't see a point paying for primary private school . I wasn't clear re what we were paying for.
In terms of "everything else", yes, private was great and I can see how social confidence and ability to speak about anything to anyone is created.