Declare-and-throw vs. throw-without-being-declared exceptions

In Java, what is the difference between the twin methods?

public void methodA() throws AnException {
    //do something
    throw new AnException();
}

public void methodA() {
    //do the same thing
    throw new AnException();
}

I have a intuition that it has something to do with being a well-designed method (because I put methodA in an interface, declared it the way methodA* does in its implementation and received a warning from Java that "A* cannot override A because A* doesn't throw AnException").

Is this speculation correct?

Is there any other subtle connotations in the two ways of doing things?