Depends on what you mean by "you". Each will experience events independently; there wouldn't be some sort of metaconsciousness shared between them. They will both be "you"s, but they will be separate "you"s, is my point. Original vs copy is theoretically valid because you can trace the material that makes up the patterns, but it matters as much as the difference between the original manuscript for a novel and a printed copy does for the information it contains. At any rate, while I'm aware that the pattern that defines my mind is constantly changing, the part of it that it can observe remains consistently recognizable as itself. There's continuity there, even if it's not identical.
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I guess all I was saying is that arguing that your brain changes all the time (and is made up of entirely different molecules eventually) misses the point, since the actual physical object isn't relevant so much as the pattern it forms.
As I said, continuity as is being presented is effectively an illusion as both new instances in this hypothetical are being created instantly. That one instance was created from a template where most of the components were already in place isn't what I would call a real difference from another instance created entirely in one moment if the information contained in each were identical. Personally I could never consider one version more real than the other, but rather would have to consider each a new sub-identity branch from the original. A single continuous identity would only be possible so long as the chain only fathered one result, and that identity would be dead once the entire chain could no longer grow.
When I referred to "You" I was referring to a point in the identity chain right before the proposed divergence, a point that would describe the entirety of a person at a particular moment. Obviously we never remain at one point for the duration of our lives, and in certain cases such as brain damage who we are can change drastically from who we were in another recent point of our identity chain. Yet our intuitive understanding of identity still tries to see such drastically different beings as the same, despite rationally recognizing that one is not the other due to dissimilar properties. The same can be said for any object that has slowly been modified, such as the object of The Ship of Theseus problem given in this thread, and rationally we know that a boat with pieces replaced due to repairs has new properties depending on the replacement, and even without being replaced we know that the boat's properties change with wear on its components. I am rather certain that the system that makes up our consciousness is the same, with property changes that are simply too small to notice in a short timescale or are irrelevant to our lives.
If Identity can exist, it is only as an abstract and, like a biological species, it will always have to be a nebulous idea which doesn't quite wrap itself around the reality of our universe but is used for convenience of comparison. The truth is that we consider ourselves to be who we are now, not the us that might exist in the future. What makes us who we consider ourselves to be is the accumulated history and the complete system that is created from our memories and other physical components. A perfect duplicate would acquire the entirety of that, and therefore be a real continuation of the self.
Here is a counter-hypothetical to the original question:
If it were possible to externally observe the process where a person is duplicated, and during the process the "original" suffered a case of complete and permanent amnesia, which of the two would be more similar to the template which created both? The one whose properties as a person better matched the original, or the one whose structure shared the original's components?
I would always say the so-called duplicate is more of the same person than the one suffering from complete memory loss. The amnesiac's only ties to the original is the state of its physical form created through the original's history, something shared by the duplicate albeit simulated instantly instead the usual method.