twig vs cotton on

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

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

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

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

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

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