From my understanding, spherical harmonics are sometimes used to approximate certain aspects of lighting (depending on the application).

For example, it seems like you can approximate the diffuse lighting cause by a directional light source on a surface point, or parts of it, by calculating the SH coefficients for all bands you're using (for whatever accuracy you desire) in the direction of the surface normal and scaling it with whatever you need to scale it with (e.g. light colored intensity, dot(n,l),etc.).

What I don't understand yet is what this is supposed to accomplish. What are the actual advantages of doing it this way as opposed to evaluating the diffuse BRDF the normal way. Do you save calculations somewhere? Is there some additional information contained in the SH representation that you can't get out of the scalar results of the normal evaluation?