cotton on vs twig

cotton on is not a noun while twig is a noun.

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

As verbs, cotton on and twig are synonyms defined as:

Other synonyms of cotton on include catch on, get it, get onto, get wise, latch on, tumble.

cotton on (noun)	twig (noun)
	a small branch or division of a branch (especially a terminal division); usually applied to branches of the current or preceding year

cotton on (verb)	twig (verb)
understand, usually after some initial difficulty	branch out in a twiglike manner
	understand, usually after some initial difficulty