Do quantum logic gates work differently than traditional logic gates?
In layman's terms, the answer is yes: quantum gates are quite different from classical gates.

One reason is that quantum gates must be reversible. This practically means that AND gates don't even exist in the quantum world (at least, not in the same sense of a classical AND gate). Instead, there is a different set of gates (e.g., CNOT, TOFFOLI, FREDKIN, etc.) that allow to do all that is possible by quantum computing, without breaking the reversibility condition.

The other issue is what you hint in your question. Qubits can be "both in 0 and 1", and the quantum gate must take care of this. But this is not a problem, it just means that the output must be a superposition of 0 and 1.
For instance, let's take a simple NOT gate. If the input is a "superposition" of 30% 0 and 70% 1, then the output becomes 70% of 0 and 30% of 1. Of course this is a very superficial explanation, and quite inaccurate, but the idea is exactly that. 

Finally, no measurement is done in gates, unless the gate is a measurement gate (also, measurement is not reversible, so it cannot happen in "standard" gates)

For more information, check


https://en.wikipedia.org/wiki/Quantum_gate
https://en.wikipedia.org/wiki/Reversible_computing