Disadvantages: Cost. Also the fact that former friends may become resentful, and bully the child who has gone from state school to private school. The private school best suited to your child may be a long way away, and your child may not be suited to boarding.
Advantages: small classes - my youngest son was in a class of only 12. Much bigger choice of subjects and sports. Lots of languages to choose from, maybe including japanese or Chinese, with possibility of exchange visits. Sports like Fives, Climbing ( school's own indoor climbing wall, for instance) Lacrosse. Usually private schools have huge playing fields. Also options for music - playing a wide range of instruments. I found teachers were much more dedicated than in state schools , much more prepared to organise after-school or vacation activities such as skiing.
And if you are class-conscious, your child will grow up with classier friends than s/he would have in a state school. A larger proportion of private school pupils reach university than state school pupils. They are better prepared, and more encouraged.
BUT it may in the past have been easier for would-be paedophiles to get employment in private schools, especially church-sponsored private schools. But we have to hope that is a thing of the past, don't we.