It depends entirely on the school in question. DD1 went to a private primary until we moved somewhere with good schools.
Academically it was brilliant- but when she got to the state primary she was indeed about two years ahead. So she spent the next few years pretty bored.
Reporting-wise it was good. They did the (Durham-?) test each year so you got an accurate picture of where your child was as against other children across the country who took the test. State has always been less informative IME.
Socially it wasn’t great, as the small class size combined with the presence of an evil little madam in her class who revelled in playing off kids against each other and excluding those who didn’t toe the line.
Extracurricular were decent, but there were limited spaces and many cost extra. State was actually much, much better in that regard.
Additional costs were better at state. Uniform for one thing costs a fortune at private and I still don’t consider there to be any need for a 6 year old to have a blazer.
SEN was seemingly good (about half the class struggled with dyslexia, dsypraxia or had issues stemming from trauma).
Ethnic diversity (though not class based diversity) was good, completely unlike the local state schools in that area at the time. But when we moved both ethnic and class diversity was better at the new state school.
Music lesson provision was far better at private (as compared to both the local state schools and the new state we moved to). Private allows for longer lessons, the teachers don’t chop and change, and there was much more musical emphasis overall.
TLDR - depends on where you are, the child’s interests, and what the local competitors are like.