Flipped signs is completely irrelevant.
An eigenvector is not unique, since you can multiply it by any constant and still have a valid eigenvector. A factor of -1 does not even impact the norm. So flipping the sign on it changes nothing, just that factor of -1.
A relative difference of 1e-14 is also irrelevant. Just floating point trash, because the computations were done in a different sequence. NEVER trust the least significant bits of a floating point number.
So, no, that is not the "interesting" part. In fact, nothing about what you have said is even remotely surprising.