Good post, Quark!
One difference is that we assume (quantum mechanics and chaotic systems aside for the moment) that on the whole, nature is deterministic--once you have a formulation for a body dropped from height X on planet Y, with known wind resistance Z, you can pretty much reproduce it. Hence mass production, engineering, etc.--in short all the "practical" things which make Western life so comfortable, in defiance of the "noble savage" ideal.
OTOH, God does what He pleases, and "reserves the right" to interfere at any time. These are called (variously) "miracles" or "mythology" according to whether we lend credence to the source, the supernatural person or agent involved, or the action which that Deity is said to have performed.
One other difference is that miracles are usually embedded in history, rather than taking place in a controlled environment subject to repetition and verification. I.e. despite the Saturday Night Live skit, we can't really re-create the Battle of Waterloo, only this time give Napoleon a B-52. Likewise, we can't go back to Jericho and have the army march around blowing their trumpets only six times; nor (as Isaac Asimov suggested) look for sappers undermining the city walls.
Cheers!
Brownian Motion (neither QM nor chaotic) seems to be another non-reproducible (in detail) system. The existence of atoms (or molecules or whatever particles) implies Brownian Motion so Einstein's work was rather interesting in this regard. The problem is that a probe that could measure the locations of (the large) particles needs to be about the size of the measured particle and is thus subject to Brownian Motion itself.