Spath seems to be taking 'public schools' to mean publicly-funded schools (which is what 'public schools' means in the USA) whereas Theland's post appears to be referring to the most elite independent schools, which are members of the Headmasters' Conference (such as Eton, Harrow, etc). In this country having 'a public school education' means having attended Eton, Harrow, Rugby, Charterhouse, Winchester College, etc.
Those schools tend to succeed because
(a) they're selective (i.e. pupils need to pass an entrance examination to get in, thus weeding out the weaker pupils) ;
(b) they've got more money to spend (e.g. pupils are far more likely to be offered rowing at Charterhouse than at most comprehensive schools) ;
(c) the parents of the pupils - or the nannies they employ - will have started formally educating their children almost from birth, rather than leaving it all to teachers later on ;
(d) the majority of public schools are boarding schools, meaning that pupils can have academic work thrown at them from early in the morning until bedtime ;
(e) public schools, like most independent schools, generally have parents who'll support the schools in their aims (particularly as they're paying tens of thousands of pounds to send their children to those schools) whereas some schools in the state sector have plenty of parents who'll tell their children not to do homework and who'll threaten to beat up teachers who try to insist that it's done ;
(f) children whose parents are paying mega-bucks for their offspring to attend a particular school are likely to have "education is important" drilled into them from the earliest age, whereas parents of pupils in state-run schools are far more likely to condone absence. (It can be hard for even the best pupils to learn in a classroom where the teacher has to devote a large part of his/her time to instructing other pupils who weren't present when key elements of the course were being covered) ;
(g) schools with more money can have smaller class sizes, making it easier for teachers to devote the necessary time to individual pupils ;
(h) pupils at public schools tend to have higher aspirations than many pupils in state-run schools (i.e. pupils in public schools expect to go to university whereas many equally able pupils living on 'sink estates' aren't encouraged to think beyond leaving school at the earliest opportunity); and
(i) there are still far too many employers (particularly in the City) who regard a public school education as automatically being better than a comprehensive school one, and who similarly regard an Oxbridge degree as automatically better than one from the University of East London, and who therefore open up career opportunities only to a privileged elite.