It's from Middle English and therefore not used today, but there are many literary and biblical uses of it, and a widely-read and educated individual (no offence intended, entangledbank!) will probably understand it.
E.g. Othello in the eponymous play (Act 5, Scene 2): "An honourable murderer, if you will; / For naught I did in hate, but all in honour."
"Naught" or "nought" (nothing) still survives (in British English, at least) as one of the words for the figure 0 (zero, nought, nothing, zilch etc) but it definitely sounds outdated.
It's even a little stronger in meaning than your teacher said - it means "it is better to have anything at all than nothing".
Unlike entangledbank I have heard it and read it in both forms (aught than naught, owt than nowt). Nowt means exactly the same as naught, but owt means anything rather than anything at all, but that is only a very small difference in meaning.
The words aught and naught are certainly archaic, whereas owt and nowt are not.
Given the post above, I should add that the spelling naught is archaic in British English, not the word - but according to the COED, that spelling is still a normal variant in American English.