What exactly makes quantum computers faster than classical computers?

"What feature of a quantum algorithm makes it better than its classical counterpart?"

First, a classical algorithm can be thought of a quantum algorithm that makes no use of quantum superpositions. Therefore a quantum algorithm can be at least as good as its classical counterpart. No classical algorithm can be "better" than quantum algorithms can do, because one example of a quantum algorithm is the classical algorithm itself.

What is a feature of quantum algorithms that make them better than classical algorithms? The feature of quantum entanglement and superposition, which allows us to answer the Deutsch-Josza problem with 1 query instead of $2^n$.

"Are quantum computers faster than classical ones in all respects?"

I believe we do not know much about "speed". We know that we can factor the number $n$ with $log^3(n)$ operations instead of $n$ operations, but how fast are those operations going to be? We only know the answer to this question for quantum computers that have < 10000 qubits, and these have not been able to factor numbers larger than about 7 digits. You can see in WolframAlpha that factoring any number with ~10 digits finishes instantly, so even if the IBM quantum computer is faster, it is insignificantly faster. We need millions of physical qubits (100s of logical qubits) to make a real comparison, and what that million-qubit architecture will look like (if it's even possible to make) is something we don't yet know. Maybe to make a device with a million qubits will come at the sacrifice of gate fidelities or gate speeds.

The answer is "no" quantum algorithms are not better in "all respects" because they are harder to implement physically!