The two theories are structurally very similar. The only difference I've noticed is that the classical theory of probability implies that the probability of an event cannot be different given two different pieces of background knowledge, i.e., that the probability of an event is a fact about the event in possibility space.

Different people may be equally undecided about different things, which suggests that Laplace is offering a subjectivist interpretation in which probabilities vary from person to person depending on contingent differences in their evidence. This is not his intention.

MSF does not hold that any event has a probability independent of some agent's evidence, probabilities are propositional attitudes, hence, properties of an agent. MSF doesn't hold that P(E) = (number of worlds where E holds) / (the number of possible worlds), because MSF doesn't think that there are non-conditional probabilities. You always assign a probability based off of your background knowledge, i.e., given the other propositions you hold.

However, most of the problems that make difficulties for the classical theory, also make difficulties for MSF theory. And MSF must either be modified in some way to address those issues, or discarded as a hypothesis.