In my understanding, the biggest difference between bump and normal map, is that a normal map actually contains 3dimensional information (in the r, b and g channel), while a bump map only has height information (black is low, white is high).

So when you wil get a 'hill' when black gradually goes to gray, a normal map can contain information about the side of the hill, apart from what you see from above...

For instance, in my 'main' 3D app Carrara, I can (with a plugin), generate a normal map from geometry (like a face). When you use a map like that, the face you see in the resulting render looks really like a complete modelled face 'rising' from the surface.

The interaction with light and shadows is much more convincing, but without the render load of displacement maps. It is used a lot in the games industry, to make low-poly models look high-poly, without displacing. It is very powerful.