Raw video is akin to shooting color negative film from the days of yore. Shooting in the standard codecs provided by Canon, Nikon, Panasonic, and even Sony to some degree used to give you very little room to play in post. The image was asis, what you see is what you get, with very little room for error, like the old slide film. Which isn't to say that all codecs are bad.
Apple's ProRes codec is one of the most robust in the industry, which is why it's the intermediate of choice when working with cameras that are forced to shoot in proprietary codecs. Some companies allow you to shoot in ProRes directly, saving you a step in post, and some (notably Blackmagic Design) don't use any proprietary codecs at all. Blackmagic Design allows their cameras to shoot in a few different flavors of ProRes (offering different levels of compression, nearly all of which are vastly superior to proprietary codecs) and different varieties of lossless raw.
That said, most companies have since come out with more robust codecs and flat picture styles and looks, in order to help bridge the gap between the highly compressed, baked in look of their codecs, and the industry standards of ProRes and raw. But what are the specific advantages of raw?
The easiest to point to is the increase in Dynamic Range. Raw images record more detail in both the highlights and the shadows, allowing you to retain information that would otherwise be lost to either pure black or pure white. This enables the cinematographer to have control over what point in which detail is lost. The cinematographer is also able to ensure a smooth roll off from the areas that are totally blown out, to the areas with detail. This is key to keeping a professional filmic look, rather than looking like video. When looking at a raw image, you may notice it looks very grey and unremarkable. Surprisingly, this dull grey quality is what gives the raw image it's immense image quality, not the least of which has to do with it's color. While the image appears grey before processing, the raw file has recorded all of the color data for each and every frame, allowing the data to be manipulated in post.
Anything that offers the cinematographer control in post is a good thing, as this means that potential problems can be fixed, and creative decisions can be made after the fact. While it's not optimal to do such things after-the fact, even things like a character's shirt color can be manipulated with ease in a raw image file.