What is the difference between subtyping and inheritance in OO programming?

In addition to the answers already given, here’s a link to an article I think is relevant.
Excerpts:

In the object-oriented framework, inheritance is usually presented as a feature that goes hand in hand with subtyping when one organizes abstract datatypes in a hierarchy of classes. However, the two are orthogonal ideas.

• Subtyping refers to compatibility of interfaces. A type B is a subtype of A if every function that can be invoked on an object of type A can also be invoked on an object of type B.
• Inheritance refers to reuse of implementations. A type B inherits from another type A if some functions for B are written in terms of functions of A.

However, subtyping and inheritance need not go hand in hand. Consider the data structure deque, a double-ended queue. A deque supports insertion and deletion at both ends, so it has four functions insert-front, delete-front, insert-rear and delete-rear. If we use just insert-rear and delete-front we get a normal queue. On the other hand, if we use just insert-front and delete-front, we get a stack. In other words, we can implement queues and stacks in terms of deques, so as datatypes, Stack and Queue inherit from Deque. On the other hand, neither Stack nor Queue are subtypes of Deque since they do not support all the functions provided by Deque. In fact, in this case, Deque is a subtype of both Stack and Queue!

I think that Java, C++, C# and their ilk have contributed to the confusion, as already noted, by the fact that they consolidate both ideas into a single class hierarchy. However, I think the example given above does justice to the ideas in a rather language-agnostic way. I’m sure others can give more examples.