In classical computation, a bit can have the value of either $1$ or $0$ and one can apply a logic gate to this bit. As far as I understand, in classical computation, no matter what gate is used, the value must still be either $1$ or $0$.
I understand that before measurement, a qubit can exist in a superposition of states, but that once it has been measured, it too must either produce the value of $1$ or $0$. However, it appears that there are many quantum logic gates that will operate on a qubit and produce a value that is not necessarily $1$ or $0$ (i.e. it the Z gate will change the value of $1$ to $-1$).
So, are we simply changing the basis? Furthermore what is the advantage or having all of these values that a qubit can be in, rather than simply the binary?