Your thought experiment doesn't really prove anything except that you can have two identical beings in the same room and they can be aware they are made of different atoms. But they are still the same person.
Perhaps you are hung up over the fact they diverge by a few seconds of experience? I would trade losing some memories for immortality. People take drugs or suffer from brain damage which has similar effects and I don't consider them "dead". In any case making a copy of the person is likely to be a destructive process, so that isn't even an issue (there's no reason to keep your physical brain after you upload it to a computer.)
Making a copy of the person isn't any different than if I just took the person and replaced all the atoms in their brain instantly with different ones. Hell this could be happening to you every second and you would have no idea. (In our universe, at least, it isn't even possible, since atoms are all identical and don't have little tags saying "atom 1250". I'm just trying to give an idea why basing your identity on a physical object makes no sense.)
If you don't consider your experiences and brain state to be you, and you don't consider the physical object of your brain to be you, then I have no idea how you define personal identity.
http://lesswrong.com/lw/pm/identity_isnt_in_specific_atoms/