the main difference between bump map and parallax displacement map is that the parallax will actually nudge pixels of a texture to occlude itself (or *overlap* itself) to create the illusion of depth.

If you have a texture of a brick wall, with all the grout between each brick and apply that texture to a plane with a bump map, you will never see a brick's SIMULATED extrusion actually cover the grout that should be behind it. Parallax displacement will, greatly improving the illusion of depth.

Here's a good comparison:
https://www.pcgamemods.com/core/6739/screenshots/b1.jpg
https://screenshots.filesnetwork.com/...s2/49663_1.jpg

I want to stress that it's the *illusion* of depth... the depth is never actually there.

Here's a big technical pdf file that explains it better than I have the vocabulary for:
https://www.infiscape.com/doc/parallax_mapping.pdf