With the limited Java knowledge that I have there is literally not a thing different about them than a abstract base class in C++. If you don't completely implement an interface in Java, then the thing that you implemented is an abstract class, and neither interfaces or abstract classes can be instantiated. The previous sentence applies exactly to abstract base classes in C++. Maybe there is a real, tangible difference between inheritance and interfaces that I'm not getting from your post but I am comfortable with the idea that this is Java's syntactic sugar at work.
If you don't know any java, I guess you could look at an interface as as sort of abstract base class. Hence, you could (I think) emulate a java interface in C++, but you would (probably) use virtual functions, which of course do exploit inheritance.
But a java interface is not actually a class, and the relationship between an interface and a class is about implementation, not inheritance.