In fact, from what I know, is that height maps are used by geometry shaders which output deformed mesh on the viewport (=your graphics card output).
Normal maps are used for the computation of light rays being reflected on surfaces which leaves the actual geometry mesh unchanged. However, encoding some extreme angles in the normal maps allows the lighting computation to produce “fake” bumpiness on a material. Did you ever look very closely on a bump map surfaces in some games and then rotated the view to the side? Then you’ll discover the surface is actually still flat, but just overlayed with dark parts that mimic bump structures.
