American English uses draft for all senses. British English splits them: draught for beer on tap, a current of cold air, and a draught animal; draft for a preliminary version of a document, a bank draft, and military conscription. Same word, divided by convention.
The core difference
One word, two spellings, and the split exists only in British English.
- draught (British) — beer on tap, a cold current of air, a draught horse, the game of draughts.
- draft (both) — a preliminary version, a bank draft, conscription.
- draft (American) — all of the above senses.
The game
Draughts is the British name for the game Americans call checkers. It uses the draught spelling, from the sense of drawing or dragging a piece across the board.