Good question. The difference is how the probability changes in response to new information. Learning more about the coin won't change the probability of heads

Good question. The difference is how the probability changes in response to new information. Learning more about the coin won't change the probability of heads from 50%, but doing more exhaustive polls would have likely improved our prediction about the election. However I don't think there's as much of a difference as you think. If we learnt more about how the coin was going to be flipped then that would certainly improve our estimate of the outcome. If we found out the exact way it would be flipped we could calculate the outcome exactly. So we can't really compare the two situations quantitatively, since there's no way to match up like-for-like the information we could receive. But we can say, for each possible piece of information we could receive, how much we expect it to change our probability.