It sounds like the concept you're looking for is energy.
You're already familiar with the intrinsic property of mass, which is somehow encoded in the matter itself. A proton has a mass that is not a result of an external system, but a feature of the proton itself. A proton has the same mass inside an oxygen atom as it does as a hydrogen ion or in the interior of a neutron star. At least, in one sense - I'm coming around to that.
And you're asking something like "what's the physical difference between two identical objects when one is moving and the other isn't?"
Let me backtrack to that constant-mass proton: the proton is a composite object composed of several quarks; smaller, lighter particles with their own properties that are largely irrelevant here. A proton is composed of a particular combination of types of quarks - a neutron is composed of a different set of quarks. In both cases however, the composite particles have the interesting property that they are more massive than the sum of the constituent quarks that compose them. But if you consider mass an intrinsic property of an object there's a problem with that: we're getting some mass "free," not as part of any specific particle.
For a proton the "extra" mass comes from the kinetic energy of the quarks inside. And if you think that sounds strange, let me confirm: the energy and mass are equivalent in that energy acts as mass and mass works as a form of energy. When physicists talk about 'mass-energy,' they're not combining two separate things into one idea, they're referring to a single idea by both names it was given in the past, before we knew they were actually the same thing.
If you're okay with the concept of mass-energy I'll move on to the next problem: energy isn't an intrinsic property of objects. If I put my arm out the window of my car while I drive, my arm is at rest - no kinetic energy - with respect to my body. If my arm strikes a post that is itself at rest in the ground with no apparent kinetic energy in that frame, there will be a painful transfer of energy between the two. This is because a definition of "rest" must be tied to a particular frame of reference; it is not a universal thing.
So now we can get back to your original example of one ball resting on a table while another falls with speed onto the table. At a particular instant of time the two objects appear identical, and for all practical purposes are identical. But the systems are not identical because of differing evolutions in time, and the system is where the energy information for the individual objects is 'encoded.'