British use both check and cheque but they find reason to use each for separate meanings. Americans simply use one spelling for all meanings. Here is the etymology:
exchequer (n.) c.1300, from Anglo-Fr. escheker "a chessboard," from M.L. scaccarium (see check). Its government financial sense began under the Norman kings of England and refers to a cloth divided in squares that covered a table on which accounts of revenue were reckoned with counters. Respelled with an -x- based on the mistaken belief that it was originally a L. ex- word.
check (n.) c.1314, from O.Fr. eschequier "a check at chess," from eschec, from V.L. *scaccus, from Ar. shah, from Pers. shah "king," the principal piece in a chess game (see shah). When the king is in check a player's choices are limited. Meaning widened from chess to general sense of "adverse event, sudden stoppage" and by c.1700 to "a token used to check against loss or theft" (surviving in hat check) and "a check against forgery or alteration," which gave the modern financial use of "bank check, money draft" (first recorded 1798), probably influenced by exchequeur.
It is a pity most Americans allow themselves to be cowed by a few pompous British (including Prince Charles) who denigrate perceived changes made by Americans to the holy British English as though nothing has changed in British English for a thousand years. The dropping of "u" from many French derived words should not infuriate the British who often have less appreciation for the French than do Americans. And let us not blame it all on Webster. The British changed many French words themselves to their liking. Why did the British add a u to devoir, parleor and saveor, all French words? Why do both British and Americans have an added "o" in transferring ardeur, armure, candeur, faveur, honneur, labeur, odeur, rigneur, rumeur, splendeur, tumeur and vapeur from French to English? And what of rigour? Is it rigor mortis or rigour mortis?
In fact, some spellings and pronunciations have changed in British English since Shakespeare's time while they remain the same or nearly the same in some or all regions of American English. We have differences but, as both forms have changed over the centuries, neither SBE nor SAE is sacred or the only correct form. Even having said this I realize it is usually futile to argue against prejudice and rationalization.