What is the difference between Born–Oppenheimer approximation and Condon approximation Both Born-Oppenheimer approximation and Condon approximation kind of refer to the separation of electronic and nuclear wave function. I'm confused what exactly is their difference.
Do we need to use BOA when we derive Franck-Condon factor or dipole correlation function with Fermi's Golden Rule by default?
 A: I am less familiar with molecular physics but here is my take.
I think you are right in saying that Born-Oppenheimer approximation and Condon approximation are both referring to the separation of time-scales between nuclear wavefunction and electronic wavefunction. Loosely speaking, nuclear wavefunction does not change while the electronic wavefunction evolves. 
I believe the term Born-Oppenheimer approximation is used when you calculate the actual wavefunction of the molecule. You "freeze" the nuclear part, and calculate the energy and the wavefunction of the electron, parametrized by the frozen nuclear coordinate. When the nuclear coordinate starts to change, the electrons adiabatically follow the new molecular potential given by the new nuclear coordinate. This way, you can picture a set of molecular potentials, with x-axis as the nuclear coordinate, and the y-axis as the energy. If the molecule has vibrations (motion of the nucleus), the electrons (with the same internal state e.g. s-orbital, p-orbital, etc.) will occupy higher levels of the same molecular potential, and you can vision the vibrational wavefunction as wavefunctions bound in the molecular potential, similar to harmonic oscillator wavefunctions. 
Condon approximation (personally I never heard this term, only Franck-Condon factors, but I come from atomic physics background) is more prominent in spectroscopy, when you want to excite electrons in the molecule. Different electronic states will feel different molecular potentials. Condon approximation says nuclear coordinate is not going to change during the excitation process, i.e. the vibrational wavefunction does not change. Hence, while the internal state of the electron changes and is limited by a strict selection rule (e.g. dipole transition from s to p orbital, etc.) for the nuclear part, only the overlap between the vibrational wavefuctions matter. If the overlap is good, the vibrational level can change by any amount. 
I would say Condon approximation is an application of BOA and hence BOA is a more general concept. If there is vibronic coupling in the Hamiltonian (corrections to BOA), the Condon approximation no longer holds (electronic excitation also results in a "kick" in the nuclear coordinate, so the matrix element would involve $\langle vib_{f}|\text{"kick"}|vib_{i} \rangle$). 
A: The Condon Approximation is an approximation that is done within the Born-Oppenheimer Approximation of molecular systems. Take a look at the electronic transition dipole moment for example,
$$
 \vec \mu_{if}(R) = -e\int^\infty_\infty  \psi_i(r,R)\sum_l^{N_{\text{electrons}}}\vec r_l \psi_f(r,R) dr
$$
where $\psi_k$ is the k'th electronic adiabatic state and $r$ is the set of all electronic coordinates, $R$ are the nuclei coordinates. The full transition dipole moment is a function of the coordinates of the molecule.  The Condon approximation is now the zero order term of the function around the equilibrium coordintes,
$$
\vec \mu_{if}(R) = \vec \mu_{if}(R_{eq}) + \nabla_R \vec \mu_{if}|_{R_{eq}} \cdot (R-R_{eq})   + \dots\\
\vec \mu_{if}^{\text{Condon}} =  \vec \mu_{if}(R=R_{eq})
$$
The next term in the Taylor expansion $\nabla_R \vec \mu_{if}|_{R_{eq}} \cdot (R-R_{eq})$ is typically called Herz-Berg Teller term/contribution. In spectroscopy we often neglect the dependency of the transition dipole moment so we can pull it out of the integral over nuclear coordinates. That yields the Condon spectrum, also called Franck-Condon Spectrum.
Doing a Taylor expansion and keeping only the zero order term of a molecular coordinate dependend coupling operator is what the Condon approxamtion means.
