Can a quantum computer (theoretically) do things a classical computer (literally) can't?
No. Quantum computers cannot solve undecidable problems.  

A quantum computer can be simulated by a classical computer. So, if a quantum algorithm could solve an undecidable problem, then we could also solve it classically via simulation.  However, we already know that if a problem has no classical solution, then it must not have a quantum solution either.

Also, you said:


  that's "just" because the quantum computer is more powerful than the classical one.


This is an open problem.  We do not yet know if $BPP \subset BQP$.