If one can encounter a distinction without a difference, a distinction that is merely verbal, one can also encounter a distinction with a difference.
I might, for example, say to someone, "I didn't call you a fool, I only said you didn't know what you were doing." The aggrieved person may well reply that I have made a distinction without a difference, because in the context in which the words were spoken, to say he did not know what he was doing was equivalent to calling him a fool.
In such a case we have a distinction without a difference. If I contrast soporific with sleep-causing, or if I say "I don't want to call you a fool, but you really don't know what you're doing," I pretend to display a difference but do not succeed.