The difference is mainly accuracy. Example: two pixels in a "ramp".
Bumpmap represents them as a stair.
Normalmap tells the engine wich direction they are facing, thus represents them as ramps.
Again normal maps are really worth if for game engines as people keep zooming in and normal maps work better, while at still rendering zooming in a lot isn't that usual.
Both still show artifacts at subtle bumpiness levels, but that is minimized with higher bit depth images like 16bit or 32bit images