Our children go (well DS has now left) to private schools for the following reasons:
DS: Is very clever indeed, very sporty, very sociable and at about 8 started to get very bored at primary. He is also very alpha and has always landed right in the centre of that little group of alpha "in crowd" boys. We decided he needed to be intellectually and physically stretched and if he was going to put down his roots with the alpha crowd then he would be in better company in the local private school than the local comprehensive.
DD: Is top average, quiet and very musical although totally unsporty. We actuallly sent her in Y7 to a top performing girls school with an enviable reputation. With a change of head there was a change in expectations and a significant downward shift in behaviour which was dreadful and very disruptive and dd found this very upsetting. The school was not prepared to deal with the behaviour on the basis that the girls behaving badly had other "problems". We moved her after the end of Y8 which is a natural time to move in the indy sector largely because she was not coping with the environment. We made a dreadful mistake with DD and I don't know what we would have done if we hadn't had the money to pull her out because she became terribly unhappy and there were no other state options.
Add to that the fact that we think education is very important and wanted to ensure that our children had access to a choice of modern foreign languages, three sciences and a classical language and that is rare to find where we live and that is the reason why. Other reasons are more specialist teaching between the ages of 7-11, ie, Maths is taught by a maths teacher, English, French, Science, History, etc.; the fact that poor teaching performance is dealt with quickly and parents are listened to more readily.
DD's school is quite small and nurturing but DS's was huge in spite of the fees. Bear in mind also that we are in inner London and I'm not sure London choices can be compared always to those elsewhere where populations might be more homogenous and more stable.
We actually tried very hard to make the state system work for our children and were delighted with the primary school they attended although it did not provide enough stretch for DS. We also learnt very early on that children, even from the same pod, are very different and one size does not fit all.
Ultimately, however, and not wanting to upset anybody, we had the choice because we have the money and could easily afford that choice. If we had not been able to pay the fees we would have tutored DS a bit; probably got him into a grammar such as Tiffin or Sutton and/or we would have moved out a bit to an area where the schools were better. I do honestly think that if you can't be sure of affording the fees a child's life is better enriched by nice holidays, some tutoring to fill in the gaps, and a generally comfortable life than by living like church mice.
Final point - our DC are both linguists - one ancient; one modern and this is a curriculum area which really is no longer being met, in my experience, in state schools - certainly not where we live.