I've worked in both - it's impossible quantity "better", very dependent on both the school and the child.
Assuming you're talking nice but unknown day schools, and not big famous public schools such as Eton / Harrow / Stowe, which are a totally different kettle of fish, then I would say this is the gist of it (purely my experience, others may disagree) -
Private school pros - smaller classes, usually much better facilities, usually a big focus on sport which particularly suits boys (not a pro if your child isn't sporty mind!), better behaviour / less disruption of learning, less chance of an easily led child falling in with the "wrong crowd", better extra curriculars (although some state schools are great at this but it's very variable
State school pros - honestly on the whole better quality of teaching because the oversight is so much better, on the whole bright kids who do well at state school seem to find the transition to university MUCH easier than their private school peers because they've not been spoon fed anywhere near as much, many universities also now give private school students higher offers so it's easier to get into a good uni from a state school, for significant SEN needs the support is much better in state schools
If it were me (and I had endless pots of money to choose), assuming your local state school is good (look at the recent results, not just Ofsted, and the progress 8 score - anything above 0 means that children do better than expected based on their predictions from their y6 SATs, anything below 0 means they do worse. Very close to 0 is fine), I would send an academic or significantly SEN child to a good state school. I would send a "middle of the road" child or one with a mild SEN needs such as dyslexia, who might be left behind in a big state school class, to private school. If money was tighter I would send to state school and get a maths and English tutor from the outset, not wait until they were struggling to get the tutor.