RGB assumes all light sources consist of three RGB lights, where the brightness of red, green, and blue varies. E.g. a yellow light would always be a red and a green light.
In contrast, spectral rendering allows light sources with arbitrary spectra. A pure yellow light (~580 nm) is different from a red+green light.
The physical difference is this: If you shine, for example, a pure yellow light on a scene, everything looks yellow, just more or less dark. But if you shine a red+green (impure yellow) light on a scene, green objects will be green and red objects will be red. Not everything will appear as a shade of yellow. Conventional RGB rendering can only model the latter case.
This means some light sources, like high-pressure sodium lamps, cannot be accurately rendered with RGB rendering: red and green surfaces would look too bright.
(Also note that the linked post has also a part 1 and 3, accessible via "next/previous post" at the bottom.)