flight of stairs vs stairway

flight of stairs and stairway both are nouns.

Word	Noun	Adjective	Verb	Adverb
flight of stairs	Yes	No	No	No
stairway	Yes	No	No	No

As nouns, stairway is a hypernym of flight of stairs; that is, stairway is a word with a broader meaning than flight of stairs:

flight of stairs: a stairway (set of steps) between one floor or landing and the next
stairway: a way of access (upward and downward) consisting of a set of steps

Other hypernyms of flight of stairs include staircase.

flight of stairs (noun)	stairway (noun)
a stairway (set of steps) between one floor or landing and the next	a way of access (upward and downward) consisting of a set of steps