What "you" are changes with time (consider that you're quite different now from when you were, say, 5). The implication of this is that once a copy is made, new experiences are formed by both copies and their patterns change in divergent ways. If you destroy one after a copy is made, the changes undergone by the destroyed one after the copying aren't transferred to or recreated again, and so are lost. Or in other words, if you make a perfect copy, they're identical at the moment of creation, but virtually immediately afterwards won't be. If you destroy the original before making the copy, then the copy is identical to the original at the moment it is destroyed, ideally, and so the same state last experienced is re-achieved and can develop further.

I'm struggling to think of the proper words to explain my thoughts on this subject, so I'm sure my responses about it are somewhat confusing, and my attempts to elaborate make them fairly long, I'm sorry about that.

Something that I think might be a source of some of that confusion is that I get the impression that many think of consciousness as a distinct nonphysical "thing" that is somehow tethered to the brain, such that the destruction of the brain results in the severing of that connection in a way that means it can't be caught and pulled back again by any physical process, similar to how people that believe in souls posit them to behave.

I do not believe consciousness works this way. I think that it literally is a specific form of information, or perhaps an emergent effect of certain kinds of information processing, and thus, is a part of the physical universe in the same sort of way that a digital image is (the image itself isn't "made" of any substance and can be encoded into any form of matter that can be organized into a sufficiently complex arrangement, but that organization physically exists, changing it changes the image, or produces a new but similar one depending on how you define it, and it cannot exist if no matter exists in an arrangement that can encode it), and as a result of that, getting it back just requires getting some matter into an arrangement that encodes it again. The tricky bit is that unlike a digital image, it isn't a static sort of information but a changing one. So, to take the analogy further, replace the image with a computer program that takes inputs from the world around it, and then rewrites it's own code in response to those inputs. If you take this algorithm, pause it, copy it's state and destroy the original machine while rewriting that state into a new machine in a new location, and unpause the program on the new machine, you'd get the same results as if you had just paused it, moved the original machine to the new location and unpaused it at the same time you would have unpaused the copied program. There's no basis to say that you have a different program, because they have the same code and are behaving the same. But if you unpaused the original machine, its instance of the program will change itself, and then if you destroy it now, the copy won't reach the state that that last version of the original would have reached had you brought it to the new location too. In this analogy, killing a person is equivalent to one of these programs reaching a state that is no longer continued, so if you continue it later, somewhere else, even on new hardware, that's fine, and if you create a branch and keep both running, that's also fine, but if you create a branch, and then destroy one without recording it's state to recreate it later, or just never actually run it again on a new machine, that branch has reached an end state that doesn't continue changing itself, and so you've had "someone" die.

The bit about things being made of different subatomic particles is interesting, because its actually, to my knowledge, difficult to truly prove that, because fundamental enough particles dont seem to have a lot of the differences seen between similar objects of larger scale. There are even ideas (not proven ones mind, just food for thought) that some of them might actually be the same, for example, theres an idea that there might be just one electron in the universe that bounces around in time and space such as to look like there are more of it: https://en.wikipedia.org/wiki/One-electron_universe

My point there though, is if "different" fundamental particles are so similar as for it to not even be clear that they are necessarily different, what would the underlying mechanism be for a notable difference to using different ones?

Im not really convinced by your final bit about continuity, but I think its more down to my difficulty in explaining what I mean exactly by calling it not real. I dont mean to say that we cant define a label for an idea like "no atoms entered or left this battery pack", but rather that theres no particular indication that the universe "cares" (cares really isnt the right word but I simply cant think of the right one and I guess cares is as close as I can think of, just strip the part out that implies conscious intent or thinking) about that label once we've defined it.

If we return to your analogy of cars, "cars" also arent really real, not in the sense that the concept applies only to things that are completely fictional the way, say "vampires" does, but in a sense like, there is a fundamental, non-arbitrary difference between, say, an electron and a photon, such that they interact with the physical laws of the universe in a distinct way. A car meanwhile, is just a collection of these fundamental particles, which does not have any distinct rules for itself among the physical laws of the universe, and rather has behavior that is merely emergent from its constituent parts following the behavior of those particles. The universe has no distinct concept that a given mass is a "car", but does seem to for an electron (again, "concept" isnt really the right word because it implies thinking and intent, which Im not trying to ascribe to the universe here, but again I struggle to find a word that better fits the idea that Im trying to communicate).

If, suddenly, every electron, proton etc (fundamental particles that is) in your phone's battery were suddenly swapped with others of the same type of particle from elsewhere in the universe, there would be absolutely no way to detect it. Presumably, this would break the continuity of that battery, but if we took a snapshot of the universe right before the swap, and one immediately after such that no time has passed between them, the only way there could be any difference at all between them would be if there was some kind of unique "label" for each particle fundamental particle to make them distinct from one another, something that, as far as I am aware, there isnt any evidence to suggest is the case. Without that added layer of complexity added to the universe, the swap would be like swapping one pixel of an image with another pixel elsewhere in the image that has the exact same color value- the result there wouldnt be a new image, because no information has been changed, it would just be the exact same image again. That is to say, the particle swap wouldnt be physically meaningful at all, unless you assume the universe has that specific unproven property added to make fundamental particles non-interchangeable, which occams razor would suggest I discount until proven otherwise, because a universe with non-unique fundamental particles is simpler than one with extra information to distinguish each. And if that swap isnt physically meaningful, then the universe before and after the swap dont have any change in information to them that could represent the break in continuity in the first place, which drives me to the conclusion that either continuity somehow exists outside the universe, which again adds another unneeded bit of complexity to reality that I can discount as less likely with occam's razor, or else that the concept of continuity is just one of the many made-up concepts that we use to help make the universe easier to think about, like labeling some arrangements of matter "cars" based on their general emergent properties, that dont have any true basis in the physical laws that actually describe the behavior of the universe.