I heard the newsreader on the BBC this morning, who was reporting on the Queen's admission to hospital, say 'the Queen's physician'. In Britain we don't generally use the word physician - we say doctor. However, physician is used in formal and historical contexts - for instance, there is the Royal College of Physicians, and the term Physician to the Queen is a historical, formal title. Another term for the latter is Head of the Medical Household. The position is currently occupied by Professor John Cunningham.
Physician is an older word in English, according to the OED. Its first reported usage was in the early 13th century, when it was spelled with an F (or sometimes a V). It comes ultimately from the Latin word for remedy, or medical or natural science.
Doctor was first used in the 14th century to mean medical practitioner. Before that it meant teacher or instructor. Indeed, it comes from the Latin docere, to teach. Even now, doctor has a wider application than medical contexts, referring to learned people with higher degrees in any subject.