If someone speaks prophetically, they speak as if they can see what is coming — and later it turns out they were right. The adverb sits on the saying, the writing, or the warning, not only on the person. A song can sound prophetically sad years after it was written, once the sad thing has happened. Though actually, in religious writing it can also mean in the manner of a prophet, even before anyone checks the future. In newspapers it is almost always the and-then-it-happened use. It is formal. You will not hear it in the playground, and you do not need it for small everyday guesses.