The classical way of explaining the difference is through the example of a coin that you know is biased, but you don't know whether heads or tails is favored and by how much. What is the probability that the next toss will be heads?

Supposedly, a frequentist would say that there is an objective answer, given by the bias of the coin which also equals the proportion of heads in a long run. You just don't know what it is, the only thing you know is that it is not 1/2. A Bayesian would say by contrast that since you have no information to favor one side over the other, the probability (degree of belief) you have to assign at this point is 1/2.

This only explains the question of Frequentism vs Bayesianism as philosophical interpretations of "what probability is". The practical issue of Frequentism vs Bayesianism as concrete statistical methods is often tangled with this one in discussions, but it is really a separate matter.