How do the different models of atoms differ from each other?
First, let us say, modern atomic model was the model of Thomson, the scientist that discovered the electron and measured its specific charge `e/m` ratio. Around 1900, Thomson imaged a model of the atom having both positive and negative charges, where the negative charges are located in the atoms like the plums in a pudding, the pudding being the positive charge that occupied the entire volume of the atom. Thus the name of this model was the plum pudding model of atom.
Ten years later, Rutherford conducted its famous experiment in which alpha particles were scattered on a gold thin foil. The results of this experiment allowed Rutherford to demonstrate that the atom is more likely empty than filled with positive charge, the positive charge being located in a very small space in the center of the atom. The nucleus of the Rutherford atom contained almost of the atom mass. In this model electrons occupied the remaining empty space of the atom orbiting the nucleus. Thus the name of this model was the planetary model of atom.
Only a few years later (1913), Bohr developed the model of the atom that has his name. This model was the starting point in developing the entire quantum mechanics in physics. He proposed that only certain orbits of electrons around the nucleus are stable and thus are allowed. The energy of these orbits take quantified values (in other words the energy is an integer multiple of a fundamental energy). The Bohr model of atom did explained well the behavior of the Hydrogen atom (one proton and one electron) and of a few simple atoms near hydrogen in the periodic system, but revealed some errors in computation of emission (or absorption) energy spectra of more complicated atoms. As said above this model was named the Bohr model or the shell model of atom.
As said above, the Bohr model of atom started the quantum theory development. Another famous scientist Heisenberg stated the incertitude principle for quantum particles. Following this and the de Broglie discovery (any moving mass act also like a wave) a wave function (of probability of being in a certain space element `dr` ) was attached to the electron. This wave wave function `Psi(r)` need to obey a certain mathematical equation that was developed by Schrodinger (1926). Thus the quantum model of atom was finally developed. In this model the electron is described not by a single quantum number `n` (like in the Bohr model) but by 4 quantum numbers attached to it (`n`, `l`, `m_l` and `m_s`) . A certain set of 4 quantum numbers can be taken by only a single electron in the atom. This model still stands today.