The case is closed by now.
The case is closed now.
What exactly is the difference between the two?
Well (save for highly unusual situations - see the comments for one) only the second one is grammatical. "By now" tends to be used only in the sense of a hypothetical or conjectural statement, where whether or not the thing has happened is in question. So
He should have got back home by now.
He might have finished writing his novel by now.
Would both be valid uses of the phrase.