Didn't notice how the discussion evolved... interesting.
Let me reply the physics question above.
One big difference people easily overlook is that strictly speaking it is the amplitude, not the probability that is the fundamental object in quantum mechanics. The classic experimental setup to verify this is the double slit experiment. if you have a single slit to let electrons pass through, you observe a bell-shape histogram on the screen, which is the probabilistic distribution of the electrons. If you have two relatively close slit, you will observe something very different. There is still a bell-shape envelope, but there are lots of dark spots in the envelope. This is interference.
So the hidden variable theory is basically arguing that quantum mechanics is equivalent to some classical theory with some unknown variables. Basically it is saying that quantum mechanics is just classical mechanics with probabilities.
But it is not. Experimentally one can do the double-slit experiment and see the effect of quantum interference, which can only be described, if the wave function has not only an absolute value (probability), but also a phase.
There is also a neat construction showing that quantum interference in successive quantum measurements can have larger correlations than any classical probabilistic interpretation. That is called Bell's inequality, and if one is interested in it, it is easier to find it on Wikipedia. (Something that I cannot just recall from my mind.)
