I have zero experience of state education other than hear say. All of us were privately educated so I have a fair bit of experience of that.
I would say;
If you are aiming for a grammar avoid a private school which schools 3 - 18.
Private schools vary massively, and fees too. They aren't all the same and aren't aiming to tutor your child in the same direction. Some are based on results, these a bit like grammar schools are directional and children are expected to meet expectations. Some of the children do not are invited to leave. My school was in that camp, 6 times a year girls were assessed upon reaching, exceeding and failing to reach school's expectations and fail to reach twice in a row and parents were called in and invited to leave. Quite brutal at 6 to fail to reach expectations but for the past 30 years the school has been within the top 5 girls schools so works for some. At prep level the pressure is on parents to keep the child going, 3 or 4 reading books to get through a day. That school is extreme, the other end of the spectrum are schools which focus on pastoral care, focus on teaching children to enjoy learning, to be good humans etc Those are not the schools which tend to feed into state grammars.
Private schools as a broad brush generally have fantastic wrap around care, great facilities, lots of sport, art, lamba and music options. Lots have their own pools and tennis courts, etc In my experience lots of stimulating material and plenty of reading books for young children.
State schools I have read are tasked with ensuring uniforms are affordable. Private schools it is the opposite, in context my youngest's summer dress alone is £56 (2nd smallest size), and to accompany that she requires a blazer, school scarf, gloves a summer hat and winter one summer jacket and winter coat, winter dress, shirts. Sports outfits they vary according to summer and winter sports. Ive never added it up but it must be £1000 - 2000 on uniform per year.
School trips are equally expensive, our 8 year old sent skiing in Feb £2000, of course these are not obligatory. So make sure you budget for those expenses...
Classes do tend to be smaller than state and often the school itself. We find integration is good across the school, our children often say hello to children much older and younger. Perhaps that is the same in state schools too.
As others have said all parents are engaged with school and we have some fantastic friends now as a result of our children's school.