They all behave like a spring in some medium with a certain energy and damping through the density of the medium. So an Oscillator is also a Damper (Animation) and vice versa. Better: They all are "damped oscillators".
In Oscillator and Damper (Animation) the parameterization goes over Filtertime and CyclesPerSecond, since Energy and Damping are not so intuitive.
A Damper (Animation) is an "damped oscillator" with CyclesPerSecond=0. So the oscillating is disabled by that. You can get the same behaviour with an oscillator by setting that pin to 0.