You're right, nothing would change, and both scenarios do play out the same. The difference would be the knowing about it part. That's what changes things for me. If it currently happens that way I wouldn't be aware of it, but if I knew that it was going to happen, the act of knowing, makes it an issue, for me anyway, even if it still plays out the same as every other time.

You remember going to sleep last night. So even if you died and a copy of you was made, intuitively, subjectively, you feel that tomorrow is still you, same as yesterday was same you, even if there is some technical disconnect.