Speaking from personal experience:
I went to a private school for most of my life and can say from experience that the private school system is incredibly outdated and really isn't worth the money you pay for it in most cases.
I found that bullying was much worse since especially at the secondary school level, students spend a large amount of time away from tutors and teachers, so there are lots of opportunities for some pretty awful verbal and physical bullying. Bullying was a huge problem at mine, as there was a divide between the 'rich' and 'poor' students - kids would get bullied for not having expensive phones/games consoles/clothes/etc. If you didn't live in the 'rich' part of town, you got bullied - some kids are raised with this idea that their privileged upbringing makes them superior to others and it was awful some of the old-fashioned, racist, sexist, generally terrible narrow-minded views they were brought up with - and still held when they left at 18.
Much of the syllabus is outdated as well - who needs to do 2 hours of sports every day, unless you're training for a PE A-level or being a professional athelete? Also there is little to no chance to study any art-based subjects. I currently study Fashion in uni and there was no chance to do subjects like textiles which would have given me a huge advantage - instead I had to take adult learning classes on this later on.
I got quite bad grades studying at private school because the bullying affected me so much and I couldn't keep up with the maths and science subjects that I struggled with but was forced to take anyway. I missed all of Year 10 because the bullying situation was so bad and instead did Year 11 at a state school - I made some lovely friends, got to spend much of my time doing Art and Photography and got As and Bs instead of Ds and Es, which I was originally predicted at the private school.
Conclusion: some kids suit private school, but good state schools offer many more opportunities in terms of subject choice and qualifications available, as well as a better presence of staff so bullying is less likely to take place.