The first difference between suppose and supposed is that suppose is a verb and supposed is an adjective. Now we can begin to understand the differences in meanings. Here are three examples and an explanation of their meanings.
1) I suppose I should go.
2) I am supposed to go.
3) He is the supposed leader.
In the first example, suppose has a meaning similar to think. Suppose generally expresses a belief that lacks certainty or an opinion.
In the second example, supposed means required or obliged.
In the final example, supposed means either mistakenly believed or based on not very strong evidence.