What's different in both cases is ontology, which is a fancy way of saying how things actually are. It may help to think of how you would program a computer to simulate either universe, or for that matter it may help to reduce the computation to something much simpler, like the following two functions:

    def odd_sum(n):
        return sum(range(1, 2*int(n), 2))

    def square(n):
        m = int(n)
        return 0 if m <= 0 else m ** 2

Those two functions are mathematically indistinguishable if all you are doing is putting in various values. There happens to be a vast mathematical identity that the sum of the first n odd numbers is the n'th square number, allowing for that indistinguishability. But to claim that they're exactly the same realization of this function is obviously mistaken.

Similarly, there is a qualitative difference between a wavefunction of the universe which changes and a wavefunction of the universe which does not change. It does not matter that the two positions can be made mathematically equivalent; one is X and one is not-X, and science is permanently incapable of figuring out which it really is.