According to the way things look right now, the key difference between classical and quantum mechanics is that the first calculates trajectories of objects while the second calculates probabilities (expressed as a wave equation). Its probabilistic character doesn’t make quantum mechanics unique in itself: coin tossing is about probability too, but you don’t need quantum mechanics to explain it. What makes quantum theory so puzzling is that sometimes what we observe seems to force us to speak as though the quantum coins were both heads and tails at once.
