I don't know about use case but in various distributed computing models there are problems that are provably easier for quantum computers. Unlike the classical setting where the best we have is factoring where we don't know an efficient deterministic algorithm and various problems which experimentally seem to be faster for QC (and those results often don't last long as we get better at simulating quantum algorithms classically)
I agree that quantum computers are useful. Its quantum internet that seems pointless.
As far as i am aware, none of the problems faster on a QC are helped in anyway by quantum internet.