Great example Les!

Here's how I understand it:

The effect you get from a bump map is really just an illusion, a trick of the light. Highlights and shadows appear to be caused by projections and depressions in the surface, but the surface is actually flat. Quite often that's all you need.

A displacement map, on the other hand, really does move the surface in and out to create actual lumps and hollows. As Les's example shows, this affects the profile or silhouette of the object.

I generally try to get away with using bump first, because it is so much more efficient in terms of memory usage. I use displacement when I know bump won't look convincing, such as when the material will be seen as a silhouetted edge, or when sharp shadows will be cast onto the material from other objects.

For example, if a flag-pole cast a shadow onto Les's 2 cubes, its shadow would accurately follow the irregularities of the displaced surface, but the shadow would appear straight and smooth on the bump mapped surface, ruining the illusion of irregularities.