Hey Iveth,
So this two are similar but there's a slight difference in the way they are used.
"Find out" - discover: so it is something that you come to have knowledge of, whether it is on purpose or not. In Spanish: enterarse/descubrir.
ex: today I found out that French is a difficult language.
"Figure out" - means to work something out, it requires a thinking process or some kind of deciphering. In spanish it would be: resolver/decifrar.
ex: today I figured out how to save ink and paper at work.
:)