British English distinguishes them: metre is the unit of length and the rhythm of verse, while meter is a measuring device — a gas meter, a parking meter. American English uses meter for everything. So the British split is device versus unit; the American one does not exist.
The core difference
A unit and a device, separated in British English only.
- metre (British) — the unit of length; also poetic rhythm: iambic metre.
- meter (British) — a measuring device: electricity meter.
- meter (American) — all of the above.
The compounds
British English keeps the split in compounds too: kilometre, centimetre, millimetre take the unit spelling; speedometer, thermometer, barometer take the device spelling. That is consistent once you see the logic — measuring instruments are meters.
The same pattern elsewhere
Litre/liter, theatre/theater, centre/center all follow the British -re versus American -er pattern, from the French-derived spellings that American reform simplified.