Advantages of spherical harmonics over baked vertex lighting?
Spherical harmonics are generally used for dynamic objects in your scene, while fully-baked lighting is used for static objects. 

A typical game engine will use both. During the GI pass, all surfaces marked as static will have their global lighting fully baked - generally into a lightmap texture rather than per-vertex. At the same time, a bunch of light probes distributed throughout your scene will receive omnidirectional lighting, and this gets converted into SH. Since SH coefficients interpolate nicely, a mesh moving through your scene can pick up the nearest few light probes (generally 4), and use the weighted average to decode a good approximation of the ambient lighting at its current location.

Another use for SH is to render less-important dynamic lights in a scene with many lights in it. The most important light or two can be rendered fully dynamically per-pixel, then for each renderer, the SH coefficients of the less-important lights are calculated and added to the light probe data. This allows the lights to change in real-time, and be rendered effectively for free per-vertex in the shader, with a small CPU overhead.