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.

Interfaces manifest the object's responsibility. Functions accepting objects as parameters should work with interfaces, not type instances. That way the responsibilities and capabilities are clear to the user of the interface and the implementor.

As far as how the implementor may fulfill its interface obligation, it may use inheritance, if it truly has an IS-A or subtype relationship with the base object.