Quote - My understanding of normal maps, is they can deform an object in all 3 spacial directions (x,y,z), where as displacement maps only deform "up and down" from the mesh's normal facing.

 

Normal maps produce a shading effect similar to bump map, but cannot alter the geometry in profile; i.e. they won't affect shadows and won't alter the edge of the model, pretty much like bump map.  Displacement mapping can do both of those.

edit: incidentally normal maps are pretty difficult to create in contrast to bump or displacement maps.

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