Where do "apology" and "apologue" differ in their roots?

I've only heard "apologue" today from the Wikipedia of the boiling frog experiment: "An apologue or apolog (from the Greek ἀπόλογος, a "statement" or "account") is a brief fable or allegorical story with pointed or exaggerated details, meant to serve as a pleasant vehicle for a moral doctrine".

If "apology" also came from the Greek "apologos", at which point is it different from "apologue"?

https://redd.it/1wyc19x
@r_etymology