I can’t answer in enough details about the differences on the ground. But just wanted to warn you that in America public school means fees aren’t paid. In Britain schools were fees aren’t paid are usually called state schools or comprehensive schools. In America I think fee paying schools are called private schools. In England too, most of our fee paying schools are called private schools. BUT, quite a lot of them, including all the prestigious boarding schools and the higher acheiving day schools, are also known as public school. So, although people might use the term public education in Britain in a very broad sense to mean the state funded education, usually if you say someone is very public school it conjures up thoughs of poshness and privilege. On another note, I think that schools run by the state in UK have a national curriculum that is imposed by the government. If there is anything similar in US I don’ know – but maybe it would be done at state rather than national level?