Private schools = Independent Schools.
Public Schools = the most prestigious private boarding schools based on social cache i.e. snobbery. The refernce to "public" is because when these schools were founded 400-600 yrs ago they were meant to be for the local poor - pretty ironic given they now charge 30k a year.
In the same way that all luxury cars are cars (but the reverse is not true) all "public schools" are private schools but not necessarily vice versa. What is true is many private schools like to describe themselves as public schools as it suggests they are like the schools listed below - i.e. prestigious, beautiful buildings, extensive grounds, social cache - Tom Brown's Schooldays etc; generally being a boarding school is a pre requisite. Historically the snootiest boys public schools were (and probably still are):
Rugby
Eton
Harrow
Winchester
Charterhouse
Westminster (probably most down to earth as predominantly a day school)
Between the six of them their offspring pretty much ran the British Empire and were the template for the whole private and grammar school systems in England.
Others that are also "pukka" are Shrewsbury, Tonbridge, Radley, Stowe, Ampleforth, Oundle, Uppingham, Wellington, Marlborough, Lancing, Hailebury, Clifton, Sherborne and St Pauls. Scotland has its own pecking order.
Sorry dont know wnough about the girls schools to comment.
If you went to any of the above you can safely say you are went to a proper "public school". There are a few oddities like Oakham, Millfield and Gordonstoun which dont really fit the above categories well as they dont really "feel" like stereotypical public schools but are just as successful.
The remaining boarding schools (of which there are many) are really described to the intense irritation of their alumni as minor public schools - generally all are historically copies of Eton or Rugby (this is why most traditional private boarding schools of any note belong to either the "Eton Group" or "Rugby Group". This is all about social pecking order and not academic results - your local grammar probably does as well if not better than many of the schools mentioned above - altho it probably wont have a polo team or beagle pack...
As the above is essentially based on snobbery I tend to avoid the public school term and refer to independent schools. Its also becoming less relevant now as parents are less concerned with "where daddy went" and more interested in league tables - particualrly overseas parents who wont have a clue about the above.