Have a look at the Polycount wiki http://wiki.polycount.com/wiki/Normal_map (old but still good info).

So my understanding is that a normal map stores direction information for each pixel whereas a bump map stores height information for each pixel (height maps can be used for either bump or displacement depending on how they are connected in the material).

In practice then they add relief detail without changing the geometry. Bump maps are far easier to create and manipulate however as normal maps are essentially code - if you're not sure how they work, best to stick with bump.

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