"To figure something out" you use when you find out how to do something. Examples: I figured out how to open the door to the garage. I figured out how to cook rice, without it getting too soggy. I figured out how to take the bus to school. "To discover" you use when you come across something or someone unexpectedly. Examples: While they were studying the sky last night, they discovered a new planet. I just started a new job and discovered that I have a talent for sales. While going through the old stuff in the attic, I discovered that my father once lived in Russia.

Figure out, in my opinion, is more active in nature, that is something that we do in order to find out about something. On the contrary, discover is something more passive in nature that is something that happens on its own, as a result of which we come across something new, like information, etc. Figure out, I reckon, is more of a synonym of 'explore' which is far beyond the scope of the word discover. Example of figure out used: I am stuck in this problem; I seriously need to figure out (or explore or find out) how to solve it. Example of discover used: The police discovered (or found) an abandoned car at the river bank. or Columbus discovered America in so and so year on his journey to so and so place, etc. Thanks Syed

"Figure out" is more commonly used I think not only herein US but also U.K., "discover" is a verb that we don't use in everyday conversation as much. Here's something we'd normally say: I've finally figured it out. "I'm going to figure it out" "Let me figure this out" Discover is a bit broad

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