Which one is better in case of a phonon calculation and does DFPT also have small negative frequencies near Gamma due to the numerical noise? What are their advantages and disadvantages?
From my understanding, the main difference is that DFPT is theoretically more sound where the electric field perturbations enter the hamiltonian and thus is calculated self consistently in the electronic and geometric optimization. This allows for several physical properties to be extracted (dielectric tensor, polarizabilities, Born charges, etc.) besides the phonon frequencies.
Computational limitations arise in DFPT to be restricted to small unit cells, while finite differences allow handling much larger unit cells.
If you are a VASP user, you may recognize the tags, IBRION=5/6 for FD and LEPSILON=True for DFPT, which may tell you some more technical differences.