Normal mapping just alters the normals of the surface. Parallax mapping offsets the textures so they appear 3D. Displacement mapping actually changes the surface geometry, and can be done using shader model 3.

Parallax mapping is generally used in conjunction with normal mapping.

Parallax mapping doesnt do any lighting at all, but it does offset the texture according to the heightmap and the view angle, creating the illusion of actual 3D bumps. Normal mapping alone will always look 2d , especially from an angle or when the camera is moving.

Parallax mapping solves the flat-look problem, but also introduces new problems, like texture distortion and still doesnt look correct from acute angles.

An improved solution can be found among a group of similar effects i will call relief mapping.

Normal relief maping however, is very slow, and still not accurate. The best solution i have seen so far is called cone-step mapping. I may implement this into the next Sphere update.