I think if you search back through the threads on here you will find quite a few that address this topic, with fervent advocates on either side. One of my children has done both - in summary pretty much as Yamin said - liked the nurturing environment of an indie and the friendlier ethos, but feels the grammar is better academically. Much happier at the indie - but think sublime to ridiculous or frying pan to fire and you have our situation - ie very gentle low-key indie to high-flying cut-throat GS. If you are looking at high-flying cut-throat indie I would certainly not even entertain the idea - you can get that in the state sector. Some children maybe need a lower key, gentler environment which is provided by what some on here rather disparagingly call 'lower tier' independent schools, and I think there is a case which can be made for that for certain types of children, if you have the money. And for some parents flash sporting or musical provision is important and state schools generally can't compete there, so if that is driving you, then maybe go independent.
Knowing what I know now, some 20+ years working in and now studying education, and mother of three, there is no way I personally would pay for schooling again, unless something I can't imagine now arose - it would certainly be for pastoral rather than academic reasons, as I think many schools are such pressure cookers now that lots of children struggle to survive. But everyone has to reach a decision they are comfortable with and I know there are those who feel independent schools are academically superior. I don't.