Closures are so great. And anonymous inner classes are NOT “Good Enough”.
However, some people don’t grok closures and think that they are “just function pointers / method objects / …”
So the first crucial point about closures is that they are a block of code plus the bindings to the environment they came from. This is the formal thing that sets closures apart from function pointers, anonymous inner classes and similar techniques.
The second difference is less of a defined formal difference, but is just as important, if not more so in practice. Languages that support closures allow you to define them with very little syntax. While this might not seem an important point, I believe it’s crucial – it’s the key to make it natural to use them frequently. Look at Lisp, Smalltalk, or Ruby code and you’ll see closures all over the place – much more frequently used than the similar structures in other languages. The ability to bind to local variables is part of that, but I think the biggest reason is that the notation to use them is simple and clear.