Inheriting from a class creates dispatch problems, and there's instance variables/fields/whatever to deal with. Never mind multiple inheritance, as that gets hairy real fast. With interfaces there is no hierarchy, and all there is is a type and a data, and you either pass them around together, you monomorphize to avoid the need to pass two pointers instead of one, or you have the data point to its type in a generic way. With class hierarchies you have to have operations to cast data up and down the hierarchy, meaning trade one dispatch table for another.
this! with interfaces, you get all the benefits and none of the negatives