plague vs pain in the neck

plague and pain in the neck both are nouns.

Word	Noun	Adjective	Verb	Adverb
plague	Yes	No	Yes	No
pain in the neck	Yes	No	No	No

As nouns, pain in the neck is a hypernym of plague; that is, pain in the neck is a word with a broader meaning than plague:

plague: an annoyance
pain in the neck: something or someone that causes trouble; a source of unhappiness

Other hypernyms of plague include annoyance, bother, botheration, infliction, nuisance, pain, pain in the ass.

plague (noun)	pain in the neck (noun)
an annoyance	something or someone that causes trouble; a source of unhappiness
any large scale calamity (especially when thought to be sent by God)
a swarm of insects that attack plants
a serious (sometimes fatal) infection of rodents caused by Yersinia pestis and accidentally transmitted to humans by the bite of a flea that has bitten an infected animal
any epidemic disease with a high death rate

plague (verb)	pain in the neck (verb)
annoy continually or chronically
cause to suffer a blight