The strings are identical, but they're two different strings. The string representations of references to them differ. Are they equal or not? This is a question of semantics that you can't answer without additional context.
Indeed, if we just accept the string representation as the universal measure of equality, then ambrus is right: the empty string and undef are equal, and thus all of my arguing has been moot. However, experience tells me it is a prudent bet to assume that in most situations, undef needs to be treated differently from an empty string.