As usual, you have to also consider: what is your project’s work-flow and deployment. If you are doing “a game,” say, then it might matter whether it’s a low-end or a high-end one; whether it needs to be able to run on “an ordinary 1960’s-era television” (i.e. varying and possibly-wretched output resolution), and so on and on and on.
If you are doing work that involves “node-based” compositing and other types of things, these various types of maps are simply very-useful inputs, for you to make use of as you may. (The ruling constraint of real-time rendering is removed.) You are free to construct a “pipeline” which uses these data in whatever way you might dream up. All of these maps are subtly different; therefore, useful in their own way and even more useful (perhaps) in combination.
The “bump map,” being in effect a point-by-point “third dimension” (expressed as a distance along the normal-vector), can actually be quite a useful thing because it is expressed in terms of the model.
The “normal map,” being a pure expression of how light bounces off the surface, can be more easily “baked” ahead of time … and it typically is.
You can use that information “in the typical game-way,” or you can use the two pieces of information together. When you do that, you wind up with a polar coordinate. (“Vector + distance.”)
“Displacement map?” Sure! By actually “moving points around” (without really moving them at all), you can describe variations of how the light behaves that would be difficult, or perhaps impossible (depending on the situation…) to describe by other means. You are affecting the behavior of faces by directly manipulating vertices, in a non-destructive way. AFAIK, only a displacement-map can describe that, in those terms.
Since this is a digital computer, and these days it has a bumper-crop of both fast CPUi[/i] and fast memory, it is truly meaningless to “debate” which one is “better.” (And it’s just plain rude to snipe at other people.) Just learn about them all. And, let this thread become one in which we discuss and explain how we’ve used them. (For everyone who’s talking there could be a hundred people listening, and trying to learn something.)
Is what I blathered-on about strictly correct? Not exactly. Normal map and bump map information are used differently in different circumstances. Real-time games have very different (“time is of the essence”) computational requirements. What’s most critical to see is how the information in each type of map is different. If you are not doing a game, you can do anything.