"Expect" does not have the connotation of "working out in one's mind" like "figure" does. "Figure" has a lot to do with "thinking something through". So if is is a sentence where someone is having to do some brain work to come to a conclusion, use "figure". "Figure" can often be used in place of "think", as in "What do you figure?" But "expect" almost always has a predictive quality, looking to the future. "Figure" brings to mind "calculate", and is used in mathematics, and from this connection comes its connotation of "working out mentally". "Expect" does not have this connotation.