In what situations would you use a height/bump map? I don't see height maps having any advantages over normal maps.  It's just an extra layer that may mess up any existing normal map with details that are already present there.  Are there any advantages to height maps?  

I normally use height maps for skin pours on my characters but I find normal map still out classes it.
A height map is useful if the camera is going to be close enough to the surface to "see through" the trickery of a normal map. A normal map leaves the surface where it is, and just changes how light reflects off; but if the "eye" is close enough that the difference in height itself would have been visible, that looks super weird.

Here's a heavily subdivided plane equipped with a normal map generated from a noise texture:



And here's the same plane, with the same texture, except that this time it's treated as a height map:



Isn't the second one a thousand times more believable? The first one would be good enough if seen from a great enough distance, or if the bump-map were "weakened" so that it reflected much shallower bumps; but this close up, only the height map can be realistic.