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.
Columbus discovered America in so and so year on his journey to so and so place, etc.