Quantum mechanics describe the evolution of the state of system given the initial conditions and the hamiltonian describing it. That's exactly what the Schrodinger equation says and why it has a partial derivate of time. Usually in introductory treatments only the "time independent Schrödinger equation" is addressed, but that's a sub product on a common technique used to solve some family of differential equations.
In classical mechanics you can interpret time as another coordinate, but with so many privileges that it's a little far reached call it in the same way as space dimensions. Without the metric introduced in special relativity there is no way to justify put them in the same footing. For example, in classical mechanics you cannot rotate in a way that makes the time coordinate fall in a space coordinate, while you can do that in relativity, and it's called a boost.