powerful vs mightily

powerful is an adjective but mightily is not an adjective.

Word	Noun	Adjective	Verb	Adverb
powerful	No	Yes	No	Yes
mightily	No	No	No	Yes

As adverbs, powerful and mightily are synonyms defined as:

powerful and mightily: (Southern regional intensive) very; to a great degree

Other synonyms of powerful include mighty, right.

powerful (adjective)	mightily (adjective)
having great power or force or potency or effect
having great influence
displaying superhuman strength or power
(of a person) possessing physical strength and weight; rugged and powerful
strong enough to knock down or overwhelm

powerful (adverb)	mightily (adverb)
(Southern regional intensive) very; to a great degree	(Southern regional intensive) very; to a great degree
	powerfully or vigorously