Basically, a bump map interacts with light sources to paint a picture of a shadow on a surface, thus giving the illusion of bumpiness.  Depending on the angle of the surface and the light source(s),  the shadow pictures can become inaccurate.  For instance, on Poser figures, it's cool to use a displacement map to subtly  simulate veins on the surface of the skin.  The mesh is actually pulled by a displacement map.  But use that same map as a bump map in Bryce and one side of the figure will have veins, but the other side will probably have a negative effect and it'll look like someone gouged out little trenches in their skin!  You can work around this problem somewhat by reversing the value of the bump on the side that's incorrect, but again, you're just playing with artificial shadows.
Displacement is probably the number one thing on my wish list for Bryce.  I heard from someone, unofficially, that Bryce 7 will eventually have displacement.