admonition vs lesson

admonition and lesson both are nouns.

Word	Noun	Adjective	Verb	Adverb
admonition	Yes	No	No	No
lesson	Yes	No	No	No

As nouns, lesson is a hyponym of admonition; that is, lesson is a word with a more specific, narrower meaning than admonition:

admonition: cautionary advice about something imminent (especially imminent danger or other unpleasantness)
lesson: punishment intended as a warning to others

Other hyponyms of admonition include deterrent example, example, object lesson.

admonition (noun)	lesson (noun)
a firm rebuke	a task assigned for individual study
cautionary advice about something imminent (especially imminent danger or other unpleasantness)	a unit of instruction
	punishment intended as a warning to others
	the significance of a story or event