grasp vs cotton on

grasp is a noun but cotton on is not a noun.

Word	Noun	Adjective	Verb	Adverb
grasp	Yes	No	Yes	No
cotton on	No	No	Yes	No

As verbs, cotton on is a hyponym of grasp; that is, cotton on is a word with a more specific, narrower meaning than grasp:

Other hyponyms of grasp include figure, catch on, get it, get onto, get wise, latch on, tumble, twig, intuit, digest.

grasp (noun)	cotton on (noun)
the act of grasping
an intellectual hold or understanding
understanding of the nature or meaning or quality or magnitude of something
the limit of capability

grasp (verb)	cotton on (verb)
get the meaning of something	understand, usually after some initial difficulty
hold firmly