The two Rikers are different persons because their lifelines have diverged at a point in the past, and both have experienced events that the other hasn't....
The same is true for Spock to an extent. His last few moments were forgotten by the new Spock and that's enough to consider him a diverged and a different Spock.
That's not what I'm talking about. If you think I'm arguing that the two Spocks are the same individual, you've profoundly misread my comments, because my whole point is that they are
not the same individual, not if the new Spock's mind is simply an exact copy and nothing more.
This same question often comes up with the idea of downloading the mind into a computer. If you download your mind into a computer or a clone body or whatever and still exist afterward, then your consciousness still resides in your own brain and you do not perceive or experience what your duplicate does, no matter how exact a duplicate it is. It may think it's you, but you wouldn't share its awareness, because you still exist within your own head and not its.
Therefore, if that exact duplicate consciousness only comes into being after you die, it follows that you won't "live on" in it, that your "soul" won't jump into that new form any more than it would have if you and the duplicate had coexisted. The duplicate would perceive itself to be a direct continuation of you, and so would every external observer, but
you wouldn't. You'd simply have ceased to exist and would have no further awareness.
So if all Spock did was create a copy or map of his neurological state inside McCoy's mind, then yes, the Spock that emerged from the
fal tor pan is simply a duplicate, essentially an offspring of the original. However, if you accept the hypothesis that his meld with McCoy simply opened a channel that allowed his consciousness to transfer out of him at the moment of his death, then you could say there was a continuity of awareness and mental activity, in which case New Spock could perhaps be the same continuous individual as the original Spock -- at least insofar as any of us remains a continuous individual.
As
captrek says, the regenerated Spock did paraphrase a conversation he had with Kirk after his meld with McCoy, which could be taken as evidence for the latter possibility, that his mind didn't actually jump into McCoy's head until the moment of his death. On the other hand, I'm pretty sure McCoy was a spectator to that conversation, so if this was merely a copy of Spock's mind "uploaded" during the meld, it could've gotten the memory from McCoy.