What is the difference between density functional perturbation theory (DFPT) and many-body perturbation theory (MBPT)? Also, please help me understand why approximations like PBEsol are considered as DFPT, whereas GW approximations as MBPT.
Density functional perturbation theory is a way to calculate the response of a system to some perturbation. For example, you can use DFPT to calculate the response due to small displacements of atoms (this is done in phonon calculations), or to electric or magnetic fields. These are DFT based methods, and they use a DFT ground state and have an exchange-correlation functional. PBEsol is one such functional, and you can do a DFPT calculation using the PBEsol functional, or many others.
Many body perturbation theory is a method which tries to solve for certain properties of an interacting electron system. For example, one can calculate the total energy, or the electron addition/removal spectrum. It typically involves coming up with approximation for the single particle Green’s function of the system, from which many other properties can be calculated. In most implantations, one does a DFT or Hartree-Fock calculation first, and then does MBPT based on the ground state wave functions. For example, in the GW approximation (perhaps one of the most commonly used MBPT methods), one often doesn’t do a self-consistent calculation, so that the results of the calculation depend on the DFT ground state. GW calculations based on PBE, LDA, or hybrid functional wave functions can (and often do) give different results. GW is a MBPT method since it’s a Green’s function based approach to solve the interacting electron problem.
Perhaps a source of confusion in your question is that DFPT can calculate some of the same things as MBPT. For example, you can calculate the dielectric matrix with both of them. Reading the VASP wiki might give you a sense for what sorts of physical processes are included in DFPT approaches vs MBPT approaches.
To amend AGS's answer, in density functional perturbation theory one looks at how small perturbations to the system affect properties such as dipole moments, excitation energies, etc.
Many-body perturbation theory (MBPT) is a wholly different animal. There, the idea is to treat interelectronic interactions as the perturbation, since we are only able to easily solve systems non-interacting electrons to high accuracy. An example of MBPT is Møller-Plesset perturbation theory, usually used in the second-order (MP2), which gives you an estimate for the electron correlation energy based on a self-consistent field Hartree-Fock calculation.
However, the issue in MBPT is that interelectronic interactions are not weak, even though one typically makes this assumption in the derivation of perturbation theories. For instance, the Møller-Plesset perturbation series truncated at $n$th order (MP$n$) often diverges when one goes to $n \to \infty$, see e.g. J. Chem. Phys. 112, 9736 (2000)