You still didn't tell us if you rendered with ray trace objects on or off.
My guess would be ray trace on.
The other thing is you don't show the object without displ or bump map.
Makes the guessing not easier.
The look of the last pics seems to indicate that your model might have no
normals,thus the facetted appearance.(normal smoothing doesn't work without
normals)
Judging from the pics of the maps the displacement is meant to do the big
shapes that would alter the look of the undisplaced object.This does only
work with ray trace off in TG as others mentioned already.
The bump map is for the fine detail that wouldn't cause visible changes in the
outer shape of the object.
Depending on the program you could use the normal map for the latter as well.
Bump or normal mapping as a last layer on a displaced low or mid poly model
that's a pretty common
approach.Works for many apps and game engines etc.