idea vs belief

idea and belief both are nouns.

Word	Noun	Adjective	Verb	Adverb
idea	Yes	No	No	No
belief	Yes	No	No	No

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

idea (noun)	belief (noun)
the content of cognition; the main thing you are thinking about	any cognitive content held as true
your intention; what you intend to do	a vague idea in which some confidence is placed
an approximate calculation of quantity or degree or worth
a personal view
(music) melodic subject of a musical composition