influence vs power

influence and power both are nouns.

Word	Noun	Adjective	Verb	Adverb
influence	Yes	No	Yes	No
power	Yes	No	Yes	No

As nouns, power is a hypernym of influence; that is, power is a word with a broader meaning than influence:

influence: a power to affect persons or events especially power based on prestige etc
power: possession of controlling influence

Other hypernyms of influence include powerfulness.

As nouns, power is a hypernym of influence; that is, power is a word with a broader meaning than influence:

Other hypernyms of influence include force.

influence (noun)	power (noun)
causing something without any direct or apparent effort	possession of controlling influence
a power to affect persons or events especially power based on prestige etc	physical strength
a cognitive factor that tends to have an effect on what you do	possession of the qualities (especially mental qualities) required to do something or get something done
one having power to influence another	a mathematical notation indicating the number of times a quantity is multiplied by itself
the effect of one thing (or person) on another	a state powerful enough to influence events throughout the world
	one possessing or exercising power or influence or authority
	a very wealthy or powerful businessman
	(physics) the rate of doing work; measured in watts (= joules/second)
	energy made available by the flow of electric charge through a conductor
	(of a government or government official) holding an office means being in power

influence (verb)	power (verb)
shape or influence; give direction to	supply the force or power for the functioning of
induce into action by using one's charm
have and exert influence or effect