bye vs farewell

bye and farewell both are nouns.

Word	Noun	Adjective	Verb	Adverb
bye	Yes	No	No	No
farewell	Yes	No	No	No

As nouns, farewell is a hypernym of bye; that is, farewell is a word with a broader meaning than bye:

Other hypernyms of bye include word of farewell.

bye (noun)	farewell (noun)
a farewell remark	the act of departing politely
an automatic advance to the next round in a tournament without playing an opponent	an acknowledgment or expression of goodwill at parting