It is thought the word ‘spanner’ comes from the late 18th century German word ‘spannen,’ meaning ‘to twist,’ and ‘wrench’ from the Late Old English ‘wrencan,’ meaning ‘twist,’ dating from before the Norman Conquest in 1066!
This shows that the word ‘spanner’ is newer and only came into common use well after English-speaking people had first emigrated to America, taking with them the older word, ‘wrench’.
This might be why the USA uses one word and the UK another.