Here is a good article describing (finally) a problem that only Quantum Computers can solve.

While Quantum Computing is a fascinating and important research topic, it has much more limited application possibilities than most people realize. There are almost no problems classical computers cannot solve equally well or better, irrespective of progress on quantum hardware. The two possible applications where quantum computers have a theoretical possibility to outperform classical computing are integer factoring (breaking some kinds of cryptography) and simulation of quantum states.