On the prevailing extensional interpretation of modality the difference between possibility and probability is the diffference between quality and quantity, possibility is the quality quantified by probability, see Probability Distributions Over Possible Worlds by Bacchus. This interpretation can be traced back to Leibniz's determinate possible worlds, but it became prevailing after Kripke's extensional formalization of modal semantics in late 1950s. To get probabilities one needs a positive unit measure on the set of all possible worlds, which is more general than counting because an infinity of them can be admitted. For instance, if a dart is thrown at a dartboard the possible worlds will have it sticking out of a particular point on it, of which there are infinitely many. But probabilities can still be assigned to particular regions based on their area (measure). Of course, different measures can be put on the same set of worlds, even when there are finitely many of them, so quantification is not unique.
The trouble is that in less trivial contexts possible worlds can not be precisely specified or surveyed, so they do not form a set ("sample space"). Because of the vagueness of description, or because there are too many of them, or both. Is it possible that the sun won't rise tomorrow? Depends on the meaning of possible. Is it possible that Russell could have been non-human? Depends on the philosophy of modality. Can a possible world be constructed with a level of detail even remotely approaching the actual world? Not humanly. But complete and determinate possible worlds are reasoned about by analogy with the actual world nonetheless, despite the fact that no ways of constructing and/or accessing such things are available, let alone surveying their totalities.
Kauffman, a mathematical biologist, gives an interesting analysis of the impasse this creates in biology, where the "adjacent possibles" are indeterminate:"if we do not know all the possible preadaptations that might arise in the adjacent possible of the biosphere, then not only do we not know what will happen, we do not even know what can happen! Can we make probability statements about the evolution of the biosphere by preadaptations? Consider flipping a coin 10,000 times. It will come up heads about 5000 times, with a binomial distribution. But notice that we knew ahead of time all the possibilities, all heads, all tails, and so forth. We knew the sample space of the process, so could erect a probability measure on the frequency interpretation of probabilities for this coin flipping process. But we do not know the sample space of the evolution of the biosphere by preadaptations, so can make no probability statements about it".
Probability works when it narrows possibility to a manageable set of determinate outcomes at the expense of limiting their range and making them highly schematic. Modal logic tries to do the same, but trades quantitative precision for a wider applicability of qualitative conclusions. Even that may distort the notion of possibility. As Felt writes in Impossible Worlds:
"The shadow of Parmenides seems to lie over these discussions... the anti-Parmenidean (Aristotelian) notion of potentiality, as an intrinsic character of the actual, has tended to be supplanted by possibilities (in the plural), Lewis’s “ways things could have been,” purely formal and discrete patterns. The dynamism of potentiality has been exchanged for a dust of homeless forms... Bergson was right, then, in maintaining that the “possible” (understood determinately) arises only simultaneously with the real... The actualists are therefore right in denying an independence to the possible... On the other hand, to be potentially is really a way to be, even though it is not to be actually. And this of course is just what Aristotle said in response to Parmenides, who conceived of only one way of being, being in actuality".
So to refine the short answer, probability quantifies not possibility in general, but a somewhat impoverished projection of it.