For nursery and primary, class sizes and individual care. Not that it would necessarily help with your interview, but the personnel turnover is often a selling point.
I would adjust my interview technique to the specific school set-up - if the school is very academic, I'd go into more detail there, whereas for some schools, especially at the early stages, the fact that children get to have a "homely" learning experience is exactly what parents like. We chose ds's school on the basis that the class sizes were so small that the teachers were able to spend individual time with each child, regardless of their ability. E.g., every reception child has 20mins individual reading time with a teacher every day. Also, the children were so proud of showing their little writing books - a varied range of abilities without anyone being left behind in a bigger group.
Obviously, except for the richest parents, a substantial part of families both have to work to afford these fees, so good wraparound care is essential, with good quality meals.
To summarise my feeling on schools: At primary, it needs to be a happy place. Once you love school and actually like learning, it's easier. And yes, yes, some state schools can offer the same. But OP asked specifically re private. Good luck with your interview, btw.