peeve vs vex

peeve is a noun but vex is not a noun.

Word	Noun	Adjective	Verb	Adverb
peeve	Yes	No	Yes	No
vex	No	No	Yes	No

As verbs, vex is a hypernym of peeve; that is, vex is a word with a broader meaning than peeve:

Other hypernyms of peeve include annoy, bother, chafe, devil, get at, get to, gravel, irritate, nark, nettle, rag, rile.

peeve (noun)	vex (noun)
an annoyed or irritated mood

peeve (verb)	vex (verb)
cause to be annoyed, irritated, or resentful	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