The coulomb force is not only " "smooth" depending on distance: it doesn't looks like a sinusoidal, I mean there are no strongly-marked values." In a real experiment of two macroscopic charges under an attractive potential, there will be acceleration and a continuous radiating spectrum, the two charges neutralizing each other with big sparks.
At the microscopic level commensurate with h, the planck constant, instead of the electron falling on the proton with a continuous radiative spectrum and neutralizing it, one observes discrete spectra, not predictable by Maxwell's equations.
The classical mathematical model had to be modified, first with the Bohr atom, which postulated stable orbits,still thinking classically, and then with the solutions of the schrodinger equation which developed into the theory of quantum mechanics, postulates and all.
The difference introduced with quantum mechanics is that it is all about probabilities, i.e. orbitals, and not orbits. It is a predictive theory which determines probabilities for finding a system in a specific state.These probabilities have a wave nature, manifest in the single particle at a time double slit experiment, and as far as the spectra go, the wavefunctions give the probabilities for transition from one spectral line to the other.
It is an observational fact that there are discrete energy levels, and quantum mechanics models them successfully, and predicts innumerable other possible observations correctly. Similar to the a falling apple: it is an observational fact modeled by Newtons gravitational laws which predict successfully all new possibilities of gravitational interactions in their range of validity .