I'm having trouble understanding what possibility means in modal logic. The Wikipedia page gives "It is possible that it will rain today." as an example of a statement which is possible. But in this case "possible" just means that we don't know whether it's going to rain or not.
Can't "$P$ is possible" be defined to mean "We don't know $P$ and we don't know $\lnot P$", making modal and epistemic logic equivalent? Or is there something I'm missing?
Epistemic logic is a type of modal logic. The term "modal logic" is used to refer to a gigantic class of logical systems - for example, contrast epistemic logic with temporal logic. The point is that for a wide variety of logical concepts - knowledge, time, plausibility, moral obligation, ..., there is a common flavor, and syntax, which is useful to pin down. Modal logic encompasses this whole general situation. The translation of $\Diamond$ as "is possible" is intended to be sufficiently broad to encompass all of these, so the vagueness there is deliberate.