by the bye vs incidentally

by the bye and incidentally both are adverbs.

Word	Noun	Adjective	Verb	Adverb
by the bye	No	No	No	Yes
incidentally	No	No	No	Yes

As adverbs, by the bye and incidentally are synonyms defined as:

Other synonyms of by the bye include apropos, by the way.

by the bye (adverb)	incidentally (adverb)
introducing a different topic	introducing a different topic
	of a minor or subordinate nature