neck of the woods vs vicinity

neck of the woods and vicinity both are nouns.

Word	Noun	Adjective	Verb	Adverb
neck of the woods	Yes	No	No	No
vicinity	Yes	No	No	No

As nouns, neck of the woods and vicinity are synonyms defined as:

Other synonyms of neck of the woods include locality, neighborhood, neighbourhood.

neck of the woods (noun)	vicinity (noun)
a surrounding or nearby region	a surrounding or nearby region