To confirm a point made by xlucid, the reason Public Schools are called public in the UK is because in the old days, before the government provided free schooling, the choice was between public schools (i.e. any school) or private tuition. Just as in Britain we don't call universities schools, nor do we really understand the term "home schooled", it sounds like its the home doing the learning, rather than the child...
These days, it is an exaggeration to say you can "always spot a public school pupil in the street, before you even hear them talk". That is of course unless they are wearing an expensive and posh looking school uniform. Otherwise you need to wait till you here their R.P. accent. But even then, many Public School boys and girls are apt at imitating Estuary English, from watching too many soaps.
Surprisingly, as fees increase, Public school students are becoming more diversified in their backgrounds and less easy to pick out from a crowd. The main variable now in a families ability to send their children to a fee paying school is how many children there are - at £15,000 a year a head, not many among the upper-middle classes can send more than one or two, especially when combined with an astronomical mortgage.
Public Schools in England are fast losing their traditional/cultural values. Whereas in the old days they would churn out officers for the army, professors for academia and priests for the Church, nowadays you send your child to them so they can get into a good university, and thereby a job which will let them make a lot of money. This is especially true now that most officers, professors and priests cannot afford the fees.
This is why, despite having been to one myself, I believe Public (private, independent) Schools are an impediment to democracy and should be banned. Democracy can only exist in a meritocracy, and a dual education system prevents this. The most obvious reason for the poor condition of state schools must be that government ministers rarely send their children to them, and so do not have an interest in improving them. Democracy is rule of the people. For it to work, the people need to recognise themselves as a coherant social entity. If public schools have lost their cultural value, and only serve to create divisions of wealth and identity, they should go.
Unlike darl, I do believe there is a chance of this happening. Either the government could nationalise them all, or privatise all state schools and provide parents with education vouchers for their children, which would dilute the exclusiveness of existing Public Schools. With the amount of nationalisation and privatisation that has taken place in the past 60 years (and ten years!) here in the UK, one of these can't be too far off. Perhaps there should be a vote on which? Equality Vs Choice. Could be fun.