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."

Believe it or not, these are historically the same word; in fact, there's a third variant, nought. Naught and even more so nought are considered archaic, though naught survives in a number of popular idioms, including "come to naught" (accomplish little or nothing, fail) and "all for naught" (expressing a wasted expenditure or useless effort). You may use any of these in this construction and be correct. However, in our time they do not all have exactly the same meaning, and though not is a variant of the older forms, it also has more modern meanings that are not the same. That is, there's more than one not running around, and both are applicable here. It matters not would most commonly be interpreted by modern readers to mean "It does not matter." In this case, not is an adverb modifying the verb matters, describing how 'it' matters: How does it matter? It matters not. Even if you mean it in the older sense of 'nothing,' most people will only see it as an adverb negating the sense of something mattering. It matters naught has a less ambiguous meaning, as naught can only mean 'nothing, zero, zilch'. In this sentence, naught is an adjective, describing it, and specifically the value, quanity, quality, or meaning of it: It = nothing, zero, zilch. Either is valid, but which you choose will depend on what you wish to convey.