In brief, the main difference between quantum and classical physics is the difference between a ramp and a staircase.
In classical mechanics, events (in general) are continuous, which is to say they move in smooth, orderly and predicable patterns. Projectile motion is a good example of classical mechanics. Or the colors or the rainbow, where frequencies progress continuously from red through violet. Events, in other words, proceed incrementally up a ramp.
In quantum mechanics, events (in particular) are unpredictable, which is to say "jumps" occur that involve seemingly random transitions between states: hence the term "quantum leaps". Moreover a quantum leap is an all or nothing proposition, sort of like jumping from the roof of one building onto another. You either make it or you break it! Events in the quantum world, in other words, jump from one stair to the next and are seemingly discontinuous
Electrons, for example, transition between energy levels in an atom by making quantum leaps from one level to the next. This is seen in the emission spectra, where various colors, indicative of energy level transitions made by electrons, are separated by dark areas. The dark areas represent the area through which electrons make quantum -- and therefore dis-continuous -- leaps between energy levels.
There are many other differences between quantum and classical mechanics involving, for example, explanations of the so-called "ultraviolet catastrophe", but these are too technical to discuss in detail here.
Let me just say the final difference between classical and quantum mechanics is the quantum notion of the "complementary nature of light", which states that light is BOTH a particle, which has mass, and a wave, which has none. This seemingly contradictory concept shows how weird quantum physics can be when compared to classical physics.