Since the question is about pure abstract classes then I'd say the answer is going to be related to inheritance and scope. It's something I've wondered myself many times and this is what I've come up with.
Obviously the features related to multiple inheritance have been answered previously so I won't go in to any of that. Scope is a big one though.
In an interface you can't define a member's access modifiers since they are implicitly public,...you are defining the public interface for it's eventual implementation. There's an important difference there since you can define a protected abstract member in a pure abstract class.
Inheriting from such a class definition would force the inheritor to implement the abstract member but scope it privately to consumers of the class (though it would have to be defined as protected
so unless the class was marked as sealed
further inheritors would have access).
In essence you can define a private interface using pure abstract classes. Whether that's a good idea is a different question altogether but one good use I've seen it used for is to enforce design patterns and standardize class designs.
HTH