diary vs piece of writing

diary and piece of writing both are nouns.

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

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

diary: a daily written record of (usually personal) experiences and observations
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 diary include writing, written material.

diary (noun)	piece of writing (noun)
a personal journal (as a physical object)	the work of a writer; anything expressed in letters of the alphabet (especially when considered from the point of view of style and effect)
a daily written record of (usually personal) experiences and observations