When used in conversation, I think that 'assume' a little more forceful, and a little more formal; and 'suppose' is a little more gentle and informal.
For example: imagine two friends are trying to sneak into a bar with fake identification cards. One might say to the other, in a friendly, conspiratorial way: 'Do you suppose we'll get past the doorman with these fake I.Ds?' However, 'Do you assume we'll get past the doorman with these fake I.Ds?' suggests some arrogance and annoyance.