At nearly 600 posts I expect it's all been said already but the simple answer is that a state school gets £6k pa to educate your child and a private one gets £20k+ from fee-paying parents.
It's not hard to see how they are then able to hire more staff, keep classes small, build better facilities and provide better resources.
They are often nicer working environments, with longer holidays, perks such as free lunches and fewer behaviour issues. There are downsides for staff too, but overall they don't seem to struggle to attract specialist teachers.
They do have higher standards - compulsory catch up and revision sessions, homework redone if it's shoddy.
They make some extra curriculars compulsory.
They select their pupils by ability and kick out any that don't maintain their standards.
If mainstream schools had 4-5 times the annual budget, freedom to select and kick out anybody they don't want, you'd see similar results.