get under one's skin vs vex

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

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

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

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

get under one's skin (verb)	vex (verb)
irritate	be a mystery or bewildering to
	subject to prolonged examination, discussion, or deliberation
	change the arrangement or position of
	cause annoyance in; disturb, especially by minor irritations
	disturb the peace of mind of; afflict with mental agitation or distress