no, thanks" = The answer to your offer is no, but I thank you for it.
Without the comma, it takes on a new feeling entirely:
"no thanks" = I do not thank you for the offer (and must not be accepting it, either).
But Thadeus used different punctuation. "Thanks, but no, thanks" = I thank you for the offer, but the answer is no, and I thank you for it. Again. [thankful, but redundantly so]
So "Thanks, but no thanks" = I thank you for your offer, but then again, I don't really thank you for your offer. [sarcastic or facetious]
E²
P.S. Thadeus, you say thanks twice as a matter of course?
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It is better to have honor than a good reputation.
(Reputation is what other people think about you. Honor is what you know about yourself.)