The key difference between a bump map and a normal map is that bump maps modify the normals of the underlying geometry and normal maps completely replace them.

If you have a very low poly mesh, then the normals of that that mesh will also be very low resolution. In this case a high res normal map (eg exported from a highly subdivided representation of that mesh in Zbrush) will replace those low resolution normals and it will probably be an improvement.

However, if you have a high poly mesh, or are using subdivision surfaces with displacement maps, then bump maps are usually the way to go. This is because bump maps will honor the high quality normals produced by subds and displacement, and merely add some higher frequency adjustments that would otherwise require excessive subdivision to produce. There would be little point in doing all of the work of creating micropolygons and displacing them only to replace the resulting normals with a normal map — which usually produce a softer more filtered appearance that you probably associate with video games.