There is no difference between a bump map and a displacement map, in terms of math.

However - people who know they're using bump only for bump (i.e. wrinkles and fine-grained texture, not to move the mesh) may remove some low-frequency height information, knowing that it will not help the cause, and that more bits devoted to tiny deviations would be better.

This is only motivated by 8-bit image limitations. Other image formats (like 16 bits) provide enough range to make such an optimization more effort than it's worth.

I'm referring here to overall height vs. local height variations. If, for example, you're modeling an entire face as a displacement of a spheroid, you need to deal with the entire protrusion of the nose (peak) vs.  the deepest part of the eye (valley). Since 8 bits only has 255 steps in it, total, you're going to have pretty big steps and the fine-grained details (like wrinkles on the lips) will be underrepresented - they will look like stacked pancakes.

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Renderosity forum reply notifications are wonky. If I read a follow-up in a thread, but I don't myself reply, then notifications no longer happen AT ALL on that thread. So if I seem to be ignoring a question, that's why. (Updated September 23, 2019)