power vs influence

power and influence both are nouns.

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

As nouns, influence is a hyponym of power; that is, influence is a word with a more specific, narrower meaning than power:

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

As nouns, influence is a hyponym of power; that is, influence is a word with a more specific, narrower meaning than power:

Other hyponyms of power include juggernaut, steamroller, Moloch.

power (noun)	influence (noun)
possession of controlling influence	causing something without any direct or apparent effort
physical strength	a power to affect persons or events especially power based on prestige etc
possession of the qualities (especially mental qualities) required to do something or get something done	a cognitive factor that tends to have an effect on what you do
a mathematical notation indicating the number of times a quantity is multiplied by itself	one having power to influence another
a state powerful enough to influence events throughout the world	the effect of one thing (or person) on another
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

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