I'd never heard of "duteous" but surprise surprise, there is a dictionary definition. As for usage, I see little point in using "duteous" when "dutiful" works fine. I think most people would not recognise "duteous", and would instead think you were making a joke.

Dutiful means to fulfill one's responsibilities. These responsibilities may be self-imposed. I will be dutiful to feed the dog, because I bought the dog, and it's therefore my responsibility to feed it. Duteous means to fulfill one's responsibilities willingly out of obedience. I'm feeding the dog because someone told me to, and I'm not complaining or objecting.

Dutiful means fulfill your duty. The other is either not a word or is never used. I've never heard it before and when I tried to look it up, I couldn't find a definition for it. For practical uses, I wouldn't suggest learning it. Dutiful is used as an adjective. "She is a dutiful worker."

I mean duteous