bother vs peeve

bother and peeve both are nouns.

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

As verbs, peeve is a hyponym of bother; that is, peeve is a word with a more specific, narrower meaning than bother:

bother (noun)	peeve (noun)
something or someone that causes trouble; a source of unhappiness	an annoyed or irritated mood
an angry disturbance

bother (verb)	peeve (verb)
cause annoyance in; disturb, especially by minor irritations	cause to be annoyed, irritated, or resentful
make confused or perplexed or puzzled
make nervous or agitated
intrude or enter uninvited
take the trouble to do something; concern oneself
to cause inconvenience or discomfort to