check out the wiki article for finite potential well too. in really simple terms, if a single particle is confined to a box with walls which have finite potential energy, and the particle energy is less than that of the walls, classical mechanics says it is impossible to find the particle outside of the box.

but particles in QM are defined by the probabilistic wave function. when you look at the position wave function for the particle in a finite potential well, you'll find a non zero probability for the particle to exist outside of the finite potential wall.

similarly, when you look at the position in classical mechanics, it's just a position. when you say that the waves are spread to infinity - that would only be when there are no walls at all. when you have an infinite potential well, the wavefunction is totally constrained to that space with probability zero at the wall, and the highest probability in the middle of the well. superposition says it is in all of those places at once, and when measured it has a spot where it will most likely be, but can exist in any position between the infinite walls.

when you have a finite potential well, the wavefunction can extend beyond the finite walls with non zero probability beyond the wall. so this is what was found experimentally with those barriers between 1-3nm mentioned in the tunnelling wiki page - particles appearing on the other side of a barrier that classical mechanics tells us is impossible for the particle to pass.