In design, you want the base class class A to present only an interface for its derived classes. This means, you don’t want anyone to actually instantiate an object of the base class. You only want to upcast to it (implicit upcasting, which gives you polymorphic behavior), so that its interface can be used. This is accomplished by making that class abstract using the abstract keyword. In other hand you want to use only part of functionality of class A so you create class B (as child) to reduce the coupling or implementation dependencies between systems and prevent duplicates.
But bear in mind when you define an inner class, code without inner classes is more maintainable and readable. When you access private data members of the outer class, the JDK compiler creates package-access member functions in the outer class for the inner class to access the private members. This leaves a security hole. In general we should avoid using inner classes. Use inner class only when an inner class is only relevant in the context of the outer class and/or inner class can be made private so that only outer class can access it. Inner classes are used primarily to implement helper classes like Iterators, Comparators etc which are used in the context of an outer class. About abstract class, it should be abstract to helpers, suppose your helpers should be too complicated to write abstract form for them.
In your case, I don’t remember extensive usage of nested abstract classes, maybe in Swing world.