Main advantage is that children are not penalised by their peers for being clever and working hard. It's normal. Both my two rather despise anyone who doesn't work hard, and think they're wasting their time, the school's facilities and their parents' money.
Minor advantage: teachers never have a 'doctrinal' issue with telling you exactly where your child sits in a class, performance-wise, according to assessments and tests. You always know exactly where you are and if there's a problem, they are pretty upfront in letting you know. In the state sector we could only find out quite obliquely where the children were; it seemed not quite the done thing to ask, so we didn't.
Another advantage: no silly predictions of GCSE grades when children are 12 by looking at SATs results. Some of my state school parents have had to fight for, eg, sitting triple science, being put into a higher set for English that might get them to an A because tests their children took at 11 indicated that they might not be able to manage this. Completely ridiculous as so many children, especially boys, are late developers. Also this business that used to go on of making children resit GCSEs until they get an A or making them enter GCSEs early and often. Or sitting highly modular science GCSEs. Game-playing and if it hasn't been stopped, it should be.
Main con of private is that my two have lost contact with some, not all, of their state primary friends. Though my son, now at university, spends more time with state school people than private, which pleases me.
Another con is that a few of the private school teachers aren't that brilliant. In some subject areas they don't actually have to do too much teaching, if a class is full of clever and motivated pupils; this small section of teachers can basically just set exercises in class and homework and mark them. We didn't come across too many of these, though.