manuscript vs piece of writing

manuscript and piece of writing both are nouns.

Word	Noun	Adjective	Verb	Adverb
manuscript	Yes	No	No	No
piece of writing	Yes	No	No	No

As nouns, piece of writing is a hypernym of manuscript; that is, piece of writing is a word with a broader meaning than manuscript:

manuscript: the form of a literary work submitted for publication
piece of writing: the work of a writer; anything expressed in letters of the alphabet (especially when considered from the point of view of style and effect)

Other hypernyms of manuscript include writing, written material.

manuscript (noun)	piece of writing (noun)
the form of a literary work submitted for publication	the work of a writer; anything expressed in letters of the alphabet (especially when considered from the point of view of style and effect)
handwritten book or document