The difference is in effect, mostly. Normal maps can give the illusion of depth, by means of highlighting areas of a normal map that 'pop' out along with directional shadows as determined by light sources. I like to describe ssbumps as a sort of pre-rendered ambient occlusion per-area as determined by light sources, and the given color directions of the map to create the illusion of depth.
I took this screenshot a while ago trying to figure out if I wanted normal maps or ssbumps for my world textures, and this is the result:
The ssbump is much more subtle, and the normal, while already pretty strong, is a little jarring and I ended up going with the ssbump--and also because you can use $seamless_scale, which I'll probably need a lot of inside caves in the project I'm working on.
If you ever want to know the extent of how much each kind of map is used, fly around goldrush or badwater with "mat_normalmaps 1" and take a look. I think you'll be surprised, I was. Here's just a small part of goldrush I took a screenshot of and marked all the ssbumps with a red x and all the normal maps with a blue x:
Edit: Supersandvich says seamless scale doesn't work in TF2, which if true is sad, because I haven't needed it yet, but I sure will. :*( Just like how I need refraction and selfillum to work together too. :**(