Bohr's model of the hydrogen atom was based on 'classical' physics - i.e. Newtonian Mechanics.
The circular 'orbit' of the Bohr electron was mathematically based on equating centripetal force with the electrostatic force. The model, however, assumes that, inside atoms, the accelerating electron does not radiate. You can think of this as the 'Solar system' model of the atom.
The cloud model does not work with circular orbits, rather, it relies on probability functions. No longer do we have discrete orbits, but rather a space where the electron is most likely to be. We can't know the exact location of an electron at any given time, just where it is most likely to be (this is the result of the Heisenberg Uncertainty Principle).
The most probable solution to the cloud model is equivalent to the Bohr model!
The move to quantum descriptions of the atom allowed for:
- understanding of spectra from larger, heavier, atoms
- understanding of the relative intensity of different spectral lines
- discovery of superfine and hyperfine structure (i.e. electron spin and other, smaller particles [quarks etc])
- understanding of the Zeeman effect (from spin numbers of electrons)
- a model that no longer violated the uncertainty principle (i.e. known, static Bohr orbits are not allowed)
- Didn't rely on the assumption that electrons accelerating in an atom didn't radiate, but all other accelerating electons do.
It's a huge jump, both mathematically, and in our understanding of the nature of particle physics.