I think one of the biggest applications for quantum computers which isn't well appreciated is simulation of quantum systems: chemistry and materials science. Similar to the old analog computers which were used to simulate differential equations long before they could be fully "virtualized" (simulated deterministically without noise) using digital algorithms.

There are several categories of computational problems where quantum computers will be able to solve at better computational complexity compared to classical computers. One famous example would be Shor's algorithm. It's considered to be a practical threat to many public key cryptography depending on integer factorization being hard.