peeve vs annoy

peeve is a noun but annoy is not a noun.

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

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

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

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

peeve (verb)	annoy (verb)
cause to be annoyed, irritated, or resentful	cause annoyance in; disturb, especially by minor irritations