The Cambridge Dictionary says: NAUGHT = old use or literary (also nought) nothing: (Examples) All our efforts were for naught. All their plans came to naught (= did not achieve anything). Grammarists says: Both naught and nought mean nothing, and in American English they are more or less interchangeable (though naught is the more common spelling). Elsewhere, they are different. Nought is conventionally used in British English for the number zero. In both British English and American English, naught is used in nonmathematical contexts to mean nothing, usually in the phrases "for naught" and "come to naught." "For nothing" would convey the same meaning as "for naught."

'Naught' is an antiquated word meaning 'nothing'. You will rarely hear or read it, outside of Shakespeare and other old texts. You will never need to use it yourself. The only time when this word occurs in everyday use is in British English, when it is sometimes in numbers to mean 'zero'. As a 'number', it's spelt 'nought'. For example, we sometimes refer to the decade from 2000 to 2010 as the 'noughties'.