Sleepywombat Cumfy is broadly right IMO re: teachers. In fact I think it's a huge elephant in the room in the public v private debate. Many I know think excellent teaching makes no real difference especially in the early years, all that matters is that the teachers can relate well to children. I disagree. This is because I have seen what outstanding teachers can do that have both real academic prowess AND empathy, focus and great teaching skills (which should be a given anyway really as far as I am concerned).
Sure there are fantastic teachers, bright with great subject knowledge who want to teach where they will make a difference in poorer schools etc but these are in the minority. The fact remains that a teacher with indifferent academic qualifications from a third tier university etc that wants to teach in a top 50 independent school will increasingly find they don't make the cut. If you have a 2:2 in Maths or below from a non RG university or equivalent university and want to teach Maths somewhere like Guildford High or St Pauls you are unlikely to get the job. Put simply only the best teachers have the pick of the schools.
I know something - a little - about the teacher recruitment at Brighton College for example. IMO they only take the exceptionally bright staff usually with a proven track record and all the focus and enthusiasm, people skills and empathy you'd expect as a given.
Our local comp would take you with a poor degree in Maths from an ex Poly (just giving as an example, know not all ex Polys are poor). You'd not get far with the top 50 independents with this sort of level of qualification. They are grateful for what they can get.
Teachers I know at Ibstock Place, another great school (independent and probably not even top 50 as not that selective I believe) have degrees in the subject they teach from universities like Bristol (2:1 and above) and all A or A at A'level in academic subjects - at least 6 As at GCSEs and so on. They also have exceptional skills as far as pastoral care, teaching skills and empathy go etc. If you took a magnifying glass to the teachers' qualifications at the average state school you'd not find them to be so stellar.
Does that mean that the teachers at private schools are always better? Not necessarily but the odds are stronger they will be I think and one reason parents opt to go private. Ideally I'd have a teacher who was academically outstanding, a subject specialist (where appropriate) who was kind, empathetic, charming and a great influence and mentor for my child? Isn't that something we all want? The fact remains I am more likely to get a teacher like that a good private school than an average state school.