A lot of UK expats are retirees, eg in Spain. To me the word differs to emigrant in that UK expats still consider themselves to be citizens [well subjects at least] of the UK. They wouldn't seek to be naturalised to their host country and will often spend a prolonged time in the UK (like many UK families with Pakistani heritage seem to spend a long time there). The flavour or how "expat" is used in the UK is for people on an [very] extended holiday.

People who "emigrate" from the UK primarily seem to go to Australia, by that I mean that's how the word is used. A high-school friend went to "work in the States" rather than "emigrating to America", for example.

Migrant (emigrant/immigrant) seems to be used primarily for those under some duress of compulsion, eg threat of violence or need of work that is otherwise unavailable.