Remember that exceptions are for exceptional cases. As I understand that, that happens when the error is out of your control. For example, invalid parameters are passed to a public API function, division by zero, situations like 'network connection lost', 'file not found'... this kind of things.
As a general rule, you should catch the exceptions that you know how to handle, like recovering from the error, log the error and propagate it, etc. If you don't know how to handle it, it's better to let it fail. Otherwise your application could be in an error state that you may not want.
So answering your question, it's better to be as specific as possible since every exception should be handled only if you know what to do with it (silently swallowing is a bad idea). If not just let the exception notify the user that something went wrong. Or if you want to, catch the exception to log the error and rethrow it.
There's good discussion here for C++, but the general concepts apply. I found the java tutorials on exceptions also very good.