get under one's skin vs annoy

get under one's skin and annoy both are verbs.

Word	Noun	Adjective	Verb	Adverb
get under one's skin	No	No	Yes	No
annoy	No	No	Yes	No

As verbs, annoy is a hypernym of get under one's skin; that is, annoy is a word with a broader meaning than get under one's skin:

Other hypernyms of get under one's skin include bother, chafe, devil, get at, get to, gravel, irritate, nark, nettle, rag, rile, vex.

get under one's skin (verb)	annoy (verb)
irritate	cause annoyance in; disturb, especially by minor irritations