What can PhDs offer to high school teaching that Masters graduates can't?

I refer only to academic PhDs, not doctorates in education. Many swanky fee-paying schools in England and the US hire PhDs as teachers. So what qualities are likely unique to PhDs and may not be mastered (pun intended) by teachers whose highest degree is a Masters?

Charterhouse's PhDs in math include:

Dr Graham Kemp, MSc, MMath, PhD
Dr Philip Langman, PhD
Dr Stephen Marshall, MMath, DPhil

Phillips Exeter Academy:

Zuming Feng, "Ph.D. from Johns Hopkins University with an emphasis on algebraic number theory and elliptic curves."
Panama C. Geer, M.S., Ph.D.
Filip Djordjije Sain, PhD Applied Math

St Paul's Girls' School (in London):

Damon Vosper Singleton (Head of Department) — MMath (Oxon), PhD (London)
Pip Bennett BSc, MA (Durham), PhD (Bristol)
Alexandra Randolph MMath (Oxon), PhD (Nottingham), MIMA

Tonbridge:

Head of Department
Dr Ian Jackson
MA (Hons) Mathematics : Trinity Hall, Cambridge
MMath : University of Cambridge
PhD (Radial Basis Function Methods for Multivariable Approximation) : University of Cambridge

Dr Jeremy King MA (Hons) Mathematics : St. John's College, Cambridge
PhD (Finite presentability of Lie algebras and pro- groups) : University of Cambridge

Dr Zi Wang MA (Hons) Mathematics and Statistics: Christ Church, Oxford

MMATH: Christ Church, Oxford
PhD (Sparse multivariate models for pattern detection in high-dimensional biological data): Imperial College London