A more insightful comparison starts from making statistical predictions from the classical behavior, that is, imagine that you have incomplete knowledge from the particle. Then you cannot predict the trajectory but you can still make a few inferences.
So what if you do not know the initial position of the particle? You can still know that:
i) the speed of the particle will be conserved (but the direction of the particle changes after bouncing at the borders of the box); so will its kinetic energy
ii) the probability of finding the particle in the box is uniform regardless of the speed of the particle (this one is a consequence of the other)
And how about Quantum Mechanics?
You should first know that if the box had no limits (if the particle doesn't bounce) the first two predictions would hold. However, due to the finiteness of the box:
i) there will be some uncertainty in the speed of the particle (this is the uncertainty principle in action)
ii) but the kinetic energy of the particle will still be conserved
iii) and you will not be able to find the particle with any conceivable kinetic (or total) energy: only some values are allowed. In particular the particle cannot be found with zero kinetic energy (another consequence of the uncertainty principle)
All these effects become negligible (below the measurement precision) when the size of the box increases.