DA Morgan is right. The difference is that a multiverse theory could (in principle) explain things that are (even in principle) unexplainable in single universe theories.
A multiverse theory can be defined by a probability distribution over some set of allowed universes. Assuming that this set is large enough you will always find yourself alive in conditions that are consistent with your existence.
This seems trivial, but this ''Anthropic Principle'' is sometimes used in astrophysics. However, in single universe theories it has no deeper explanation, while in generic multiverse theories it is a consequence of the theory itself.
The Anthropic Principle is actually superseeded by a more powerful principle in multiverse theories. You would expect to find yourself in the most likely circumstances. The predictive power of multiverse theories is thus greater than that of single universe theories.
