The short answer to this question is: Unfortunately, not much.
If we think of the state of a quantum program as a vector and every quantum gate corresponds to a matrix, then every quantum circuit can be simulated using basic linear algebra.
In this regard, every quantum circuit can be simulated by a classical computer. (or a slightly attentive high-school student, for that manner…)
Any computational problem that can be solved by running a quantum circuit on a quantum computer, can also be solved by simulating the circuit on a classical machine.
The only difference is the time it takes.
Runtime Complexity
When we talk about the quantum advantage of algorithms like Grover’s algorithm or Shor’s, we usually talk about the runtime complexity, i.e., how the number of operations grows as the problem size increases.
Both quantum and classical computers can solve the same problems.
In some cases it is just more efficient to solve the problem on a quantum computer, especially for large problem instances.
So, is there no difference in what each computer can compute? Is it always just about efficiency?
Almost, but not quite.
The main difference between what a quantum and a classical computer can do is randomness.
Randomness
When classical programs make use of randomness, they employ Pseudo-random number generators. Random numbers and selections look random, but they still follow a deterministic pattern.
Quantum computers, in contrast, (to the best of our understanding) realize true randomness.
During computation, a quantum computer is in a state of Superposition. The state of the computer is a probability (amplitude) distribution over all possible states.
Then, during measurement, this superposition collapses to a single value. Whereas the likelihood of measuring a state depends on its weight in the superposition, which state will be measured is truly random.
Thus is the main difference between what quantum and classical computers can do: Both can solve the same problems. Both are Turing complete. But through measurement, a quantum computer can realize true randomness.