I believe (though happy to be corrected)
Private Schools are businesses who can charge what they like and pick and choose who to allow in on whatever grounds (apart from race, religion etc)
Public schools have Charity status, can charge fees, pay virtually no tax but can't choose who attends on any other grounds than ability to pay!
in the UK -- no, nothing at all like that!
In the UK we have state schools and independent schools.
For historic reason Independent schools are commonly called private schools or public schools. ('public schools' is used especially to describe the famous, traditional posh ones).
All independent schools have charity status, but the state is keen on involving 'for-profit' companies in the running of some of its schools (the new, so-called free schools, also state funded.)
All schools in the UK are allowed to discriminate on basis of faith - in fact state schools particularly are postively encouraged to. A large proportion of state schools are so called 'faith' schools, funded entirely by the state but turned over to religious authorities to run. these schools admit children, and employ staff based on religion.
Oddly all the major parties, and most people in the UK think this is a good thing.
No schools are allowed to discriminate on grounds of race (we have got that far at least)