middle west vs Midwest

middle west and Midwest both are nouns.

Word	Noun	Adjective	Verb	Adverb
middle west	Yes	No	No	No
Midwest	Yes	No	No	No

As nouns, middle west and Midwest are synonyms defined as:

middle west and Midwest: the north central region of the United States (sometimes called the heartland or the breadbasket of America)

Other synonyms of middle west include midwestern United States.

middle west (noun)	Midwest (noun)
the north central region of the United States (sometimes called the heartland or the breadbasket of America)	the north central region of the United States (sometimes called the heartland or the breadbasket of America)