What do the Heisenberg and Schrödinger pictures in quantum mechanics correspond to in classical mechanics (if they correspond to anything)? It's kind of weird, because (if I understand it well) in classical mechanics the state of the system is determined by the dynamical variables positions and conjugate momenta, so the time dependence of the dynamical variables is the same as the time dependence of the state vector, I don't see whether the two could be separated. Nevertheless can it be done somehow?
There is really no analogue of the "pure state" or "state vector" in classical physics so there is no analogue of Schödinger's equation for ket vectors.
However, the density matrix has an analogue in classical physics, namely the probability distribution on the phase space, also called $\rho$. Of course, when we talk about it, we assume a statistical description i.e. we assume some ignorant about the classical system. The classical counterpart (and the classical limit!) of the "Schrödinger equation for the density matrix" is called the Liouville equation or the equation from the Liouville's theorem, so that it's not confused with other things called the "Liouville equation".
The quantum counterpart of the usual dynamical equations from classical physics are of course the Heisenberg equations of motion, the equations describing the evolution in the Heisenberg picture. The classical equations of motion are literally the classical, $\hbar\to 0$, limit of the Heisenberg equations of motion.
I have oversimplified a bit. There is no classical counterpart of the wave function among the things that we normally learn about. However, there exists the Koopman-vonNeumann classical mechanics
that is doing nothing else than to construct some "wave function"-based description of classical physics, including Born's rule.
From The Gateless Gate by Mumon,
The Heisenberg Picture is the the first monk, the Schroedinger picture is the second monk.
The approaches are mathamatically equivalent, you have some state space manifold and you define some time flow along the manifold. The two pictures are dual to each other, like Lagrangian and Hamiltonian mechanics (on the tangent and cotangent bundles respectively). Have a look at this paper for a more complete explanation.
The already posted answer is, strictly speaking, correct; however i will try to provide an analogy of Heisenberg/Schrödinger pictures which can be classical.
The difference between the pictures is what part of the description carries the time dependence and evolution.
Heisenberg picture, has (the operators of) the observables carry the time-evolution while the state is not affected by time evolution.
In Schrödinger picture, the state carries the time evolution, while any operator-observables are not affected by it.
Now, consider a (single particle) classical system which is evolving in some frame of reference. One can assume a picture of the system where the frame of reference is static (lets say this is "Heisenberg picture of the system") and the particle observables (position/momentum) are evolving (wrt to the frame of ref). Or, on the other hand, say the system observables are static (wrt the frame of ref) while the frame itself is evolving (lets say this is "Schrödinger picture of the system"). This can be generalised to multiple particle systems, but is good for an analogy.
If one wants a closer analogy, one can assume a system which is liquid or gas and the frame of reference can be curvlinear. This will make a probabilistic / density description possible and is closer to a "quantum-like description"