lesson vs school assignment

lesson and school assignment both are nouns.

Word	Noun	Adjective	Verb	Adverb
lesson	Yes	No	No	No
school assignment	Yes	No	No	No

As nouns, school assignment is a hypernym of lesson; that is, school assignment is a word with a broader meaning than lesson:

lesson: a task assigned for individual study
school assignment: work, such as schoolwork or piecework, that is done at home

Other hypernyms of lesson include homework, schoolwork.

lesson (noun)	school assignment (noun)
a task assigned for individual study	work, such as schoolwork or piecework, that is done at home
a unit of instruction
punishment intended as a warning to others
the significance of a story or event