I think privately educated children have 4 advantages over publicly educated ones:

1.      More funds to attract better teachers

2.      Genes: parents that have higher IQ which makes them earn more

3.      Richer parents which are giving more support and better protection, they want to see the money invested pay off

a.      Schools sometimes exploit this and involve parents in school activities

4.      Self selection: set of students that enter the school have higher average attainment at the start and less weak students pull down the average (since the public schools are deprived of these gifted children it works against public schools)

This means that it is much easier for them succeed. This is not to say that public students cannot achieve success but there are many barriers for them.