general statement what kinds of problems can be solved more efficiently
My limited understanding is that problems in which cyclic groups or their products play a central role usually have very good (exponential to polynomial complexity) speedups when quantum algorithms are used. 

This is due to the fact that the complex Fourier transform plays a central role in the representation of quantum states.

The fact that Grover's search doesn't have this impressive speedup is also due to the fact that it is not tightly related to cyclic groups.