To figure something out means you have to think to solve it. For example: You figure out a math problem. You don't find it out. You figure out how to solve a puzzle. You don't find it out. To find out something may take a little effort but not a lot of thinking. If I wanted to learn how to say something in Arabic, I could look it up and I would find it out. If I wanted to know to know the capital of France, I would find it out by looking it up.

Find out=discover some information. Figure out is usually used when we are in the process of making a decision or trying to arrange something. I can't make plans for my holidays until my boss figures out when he is taking his vacation.

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