The goal of computational chemistry is to obtain the wavefunction of a system. This is done by solving Schrodinger's/Heisenberg's/Dirac Equation. In this answer I will use the time-independent non-relativistic scenario as the default.
The Born-Oppenheimer Approximation
I am not sure what motivated Born and Oppenheimer to use this approximation. The seminal paper is in German [http://onlinelibrary.wiley.com/doi/10.1002/andp.19273892002/abstract;jsessionid=7FADE1955A99BCA84D68152228B0FBE3.f01t02]. But it appears that the motivation was to simplify the Hamiltonian. According to the introductory course at MIT,
it allows one to compute the electronic structure of a molecule without saying anything about the quantum mechanics of the nuclei
And according to wikipedia, it reduces the amount of computations
For example, the benzene molecule consists of 12 nuclei and 42 electrons. The time independent Schrödinger equation, which must be solved to obtain the energy and wavefunction of this molecule, is a partial differential eigenvalue equation in 162 variables—the spatial coordinates of the electrons and the nuclei.
As many approximations, it does not hold true in all scenarios. Cases where the Born-Oppenheimer approximation fails are [http://www.annualreviews.org/doi/abs/10.1146/annurev.physchem.49.1.125?url_ver=Z39.88-2003&rfr_dat=cr_pub%3Dpubmed&rfr_id=ori%3Arid%3Acrossref.org&journalCode=physchem&]:
ion-molecule, charge transfer, and other reactions involving obvious electronic curve crossings
Qualitatively, the Born Oppenheimer approximation says that the nucleus is so slow moving that we can assume it to be fixed when describing the behavior of electrons. Mathematically(?), the Born Oppenheimer approximation allows to treat the electrons and protons independently. This does not imply that the nucleus and electrons are independent of each other. In other words, it does not mean that the nucleus is not influenced by the motion of electrons. The nucleus still feels the motion of the electrons. In addition, the Born Oppenheimer approximation does not say that the nucleus does not move! It only means that when describing the motion of electrons, we assume that the nucleus is fixed.
No Analytical Solution to Schrodinger's Equation
Unfortunately there is no analytic solution to Schrodinger's Equation for any atom that has more than one electron even after the Born Oppenheimer Approximation (here is a list of quantum-mechanical problems that have an analytical solution: https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions). Many texts state that the reason as to why Schrodinger's equation is not exactly solvable for more than one electron is due to the Coloumbic repulsion between electrons. [http://chem.libretexts.org/Textbook_Maps/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Quantum_States_of_Atoms_and_Molecules_(Zielenksi_et_al.)/09._The_Electronic_States_of_the_Multielectron_Atoms/9.1%3A_The_Schrödinger_Equation_For_Multi-Electron_Atoms]
However, this is not entirely true. A counterargument is Hooke's atom. The Hamiltonian for Hooke's atom has an Coloumbic electron-electron repulsion term. However, it has an exact solution for more than one electron under certain circumstatnces. [https://en.wikipedia.org/wiki/Hooke%27s_atom]
The true reason as to why Schrodinger's equation is not solvable for multielectron atoms is due to the fact that the motion of electrons cannot be decoupled from each other. In other words, the Hamiltonian is not separable for a multi electron system. If we were to get rid of the electron-electron Coloumbic repulsion, the motion of the electrons can be decoupled. This may be the reason as to why the electron-electron Coloumbic repulsion (a.k.a. electron correlation) is used as the reason why Schrodinger's equation is not exactly solvable.