Be Cause?

What is a mathematical model of a physical system? At what kind of level is it really there? Is it actually there at all, or is it simply an aid to calculation?
These questions are of great importance in the philosophy of science and also to science itself, especially in the light of quantum physics and the Schrodinger Equation.
Let us take a simple model of a system - for example, Newton's Law of Gravitation:

We know it works. We know it gives true predictions. So is this an accurate representation of what is actually there? Or is this equation simply a machine that will give us a prediction but not an explanation? After all, we could represent this gravitational force in many ways. We could imagine particles - 'gravitons' that move from one body to the other. We could think of space as deformed like a rubber sheet by massive bodies, causing other bodies to roll inwards. Or perhaps there is a field created by mass, much like the one generated by charge. All of these realities could be represented by the same equation.
There may not be any way to tell apart these differing stories. After all, if they all produce the same effect, what can we possibly do to distinguish them? Are we doomed to know nothing more than our models which may turn out to be absolutely accurate in terms of effect but hopelessly lost in terms of cause?
There is a debate in computer science about algorithms, sets of instructions for the computer to follow. If two algorithms give exactly the same results in the same circumstances, are they the same algorithm? I would say 'yes'. Think about all the different ways and shortcuts there are to do multiplication. The exact same process is actually taking place here, there are simply different ways to represent it.
So perhaps all those little stories we might tell ourselves are all true at the same time. If they have the same effect, if there is no way to tell them apart, are they different at all? They may turn out to be like all the dozens of proofs of Pythagoras' Theorem - they are all exactly equivalent.
But this leaves open the question of causality. If the three ideas I have given about the origin of gravity are all true at once, what is actually causing it? Our law tells us that gravitational attraction occurs between two masses, but is that actually an explanation? Or is it just a statement of prediction - that when two masses exist, they will attract each other?
Of course, any answer to this would require us to define causality properly. We all know that just because two things are correlated, it does not mean that one causes the other. But on the other hand, if two things correlate absolutely perfectly (A always preceeds B, B is always preceeded by A), isn't that enough? If we had a perfect mathematical model (and of course we don't), doesn't that provide all the explanation we need? Or is it merely a description of the effects of the real cause?

My head is starting to hurt, so here's a cute and unrelated video.