Bump and displacement maps are the same - grayscale data is used to determine height. The difference is the way the map is used by the renderer - displacement will affect the geometry, bump will not. Displacement costs a lot more in render time than either bump or normal maps.

A bump map affects the surface normal based on height; a normal map replaces the surface normal with an RGB value that "can represent the z, y and z coordinates of the surface normal of the high-res mesh" (from Scott Spencer's Zbrush Character Creation). Bump and normal maps both fake detail. At least in theory, a normal map created from the comparison between a high and low-res version of a mesh should carry more accurate info than the bump - but I haven't yet seen a clear demonstration of this.

So when adding detail to a mesh with a map and the silhouette of the mesh needs to change, displacement is best - if not, use a bump map or, if you have the facility, try generating a normal map.

Note that normal maps come in different flavours too - object space or tangent space, plus different apps/renderers could have different requirements. Octane requires the green channel to be flipped, Poser does not - so a normal map that works in Poser wouldn't work correctly in Octane ....

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