Difference between constant of motion and conserved quantity

For a scattering process let $S$ the scattering operator and $P_0$ the energy operator of the system. We define R to be a constant of motion if

$[R,P_0] = 0$

and we define C to be a conserved quantity if

$[C,S] = 0$.

It is easy to see that there are constants of motion that are not conserved and the other way around, so those are non equivalent. I can see that the definition of ‘constant of motion’ fits the quantum mechanical picture where $P_0$ is the Hamiltonian and the motion is given by the Schroedinger equation. And clearly conserved quantity simply means that it does not change during the scattering process.

But I cannot quite understand what the difference between those two is especially concerning the physical interpretation. For me they both say that a quantity does not change as the systems evolves. Can someone point out exactly in which way they differ?

(The definitions are taken from the book "An Introduction to Symmetry and Supersymmetry in Quantum Field Theory" from Jan Lopuszánski)