Just because you don’t know which way it’s spinning doesn’t mean that there’s not some predetermined answer before somebody observes it.
I think part of why it's significant regardless is that identical Fermions (ex: electrons) can't exist in the exact same state as one another, as in you can't have 2 electrons occupy the same atomic orbital subshell with the exact same spin. If the electrons are going to be existing with one another, then they have to have opposite spins, otherwise their wavefunctions would equal zero, meaning they wouldn't exist. But for two electrons that aren't in the same state, then it doesn't matter, they can have either spin state regardless of what the other one is doing.
So by separating two entangled electrons apart, then it should follow that their spin states should be completely arbitrary relative to one another. So you should have states where both are measured to be in spin up or spin down states as well as in opposite spin states. But they're only ever measured to be in opposite states, as if they're still obeying the rules that identical fermions must follow when they try and occupy the same state, despite being far enough away from each other that those rules shouldn't apply.
(Also, I'm just an undergrad and we really haven't covered entanglement, so this is just my rough, and quite possibly wrong, interpretation of why it's important.)
TL:DR: I guess the significance is that while the act of entangling two electrons will necessarily cause them to adopt spins opposite from one another, once you separate them out the relationship that caused them to adopt those opposite spins shouldn't apply anymore. But measurements show that they continue to obey that relationship right up until they're measured.
Edit: Made sure to specify that the rules for Fermions only apply when they're identical/indistinguishable from one another.
Heh, to be honest I can't remember a lot of that! But you're right, Bell's Theorem isn't the only expalantion for why entanglement is "spooky", since Einstein said that long before the theorem even existed. But I think it's the simplest approach for explaining to a lay audience without a lot of extra ideas or "you just gotta believe me on this bit." And I think it's what definitively killed local hidden variable theories.
I think part of why it's significant regardless is that identical Fermions (ex: electrons) can't exist in the exact same state as one another, as in you can't have 2 electrons occupy the same atomic orbital subshell with the exact same spin. If the electrons are going to be existing with one another, then they have to have opposite spins, otherwise their wavefunctions would equal zero, meaning they wouldn't exist. But for two electrons that aren't in the same state, then it doesn't matter, they can have either spin state regardless of what the other one is doing.
So by separating two entangled electrons apart, then it should follow that their spin states should be completely arbitrary relative to one another. So you should have states where both are measured to be in spin up or spin down states as well as in opposite spin states. But they're only ever measured to be in opposite states, as if they're still obeying the rules that identical fermions must follow when they try and occupy the same state, despite being far enough away from each other that those rules shouldn't apply.
(Also, I'm just an undergrad and we really haven't covered entanglement, so this is just my rough, and quite possibly wrong, interpretation of why it's important.)
TL:DR: I guess the significance is that while the act of entangling two electrons will necessarily cause them to adopt spins opposite from one another, once you separate them out the relationship that caused them to adopt those opposite spins shouldn't apply anymore. But measurements show that they continue to obey that relationship right up until they're measured.
Edit: Made sure to specify that the rules for Fermions only apply when they're identical/indistinguishable from one another.
Heh, to be honest I can't remember a lot of that! But you're right, Bell's Theorem isn't the only expalantion for why entanglement is "spooky", since Einstein said that long before the theorem even existed. But I think it's the simplest approach for explaining to a lay audience without a lot of extra ideas or "you just gotta believe me on this bit." And I think it's what definitively killed local hidden variable theories.