"find out" means to learn something from a person or persons or other source(s).
'I have to find out what my work schedule for the next month is.'
(you will ask the person that decides the work schedules where you work)
"figure out" means that you will consider something and maybe gather infromation, and then you will decide what the answer is.
'I have to figure out what I will study in university.' (you have to consider this yourself; probably you will talk to others and gather information, but then you will have to decide what to study)