Both terms are quite general with borad uses.
PBR
may be considered as volume rendering, since most of the time 3D-volumes are being rendered.
Volume Rendering
may be (but doesn't have to be) physically based. Simple raycasting combined with Blinn-Phong shading for example may be an effective tool to get an approximation that is not physically based. PBR simply means that the interaction of rays with the dataset is somewhat physically based.
Volume rendering is often used synonymous to Direct Volume Rendering
(DVR). When comparing PBR to DVR one could say that mostly PBR works on objects with surfaces. Rays interact with the surfaces (scattering, refraction etc.). While DVR implementations may use similar effects, many use-cases work on voxel-based datasets. A popular example would digitally reconstructed radiographs
(DRR) where x-ray images are synthesized from ct/mri scans: Rays go through the entire model and do not stop on the first non-void object. That specific case could be implemented in a PBR manner, since you model rays interacting with the densities of you volume.