Yes, the two will diverge. But then, so they would even without rounding error, on account of quantum mechanics.

Neither of them is "the original person" (you are not, now, quite the same person as you were a year ago). Both are, so to speak, descendants of the original person. There are many (apparently) possible descendants -- you never know what might happen to you, after all. A good enough simulation would be as much like one of those as they are like one another. (I suppose that's a definition of "good enough".)