My personal experience, I went to state schools until 13. I was heading the wrong way, in the wrong crowds, getting into trouble and not interested in school or learning in the slightest.
I discussed the situation with my parents and I was lucky enough to get into a local independent school in the last year that the government offered assistant places and was also offered a part scholarship.
The school slowly but surely transformed me. It was not an instant fix, but the teachers and staff were more willing and able to spend time and by the time I entered 6th form I was loving it.
I would say that as a broad generalisation the teachers are better in independent schools. The pay and conditions are usually better, which attracts the best teachers.
The options for sport & arts was far superior, although this is only comparing 2 particular schools. Instead of having the choice of rugby, rugby or rugby I was able to pick and choose from anything I wanted to do and pupils were able to start new clubs or sports if they wanted.
The decision is very much dependant on the individual person and the area, but if I am able to put any children we have through an independent education, then I will bend over backwards to make it a possibility. The one caveat is that I wouldn’t let them board until 13 at the earliest if boarding becomes the only option.