Well, you get all the real advantages of real CSS. The browser tools work. Various other preprocessors work. You can use timesavers like Sass or Less or whatever else.

The biggest advantage, imo, of using a component based design, where components handle not only the styling but the entire appearance of a "thing", is that you make the contracts for what that thing has to support explicit. If your button needs to be able to change colors, you can add a prop that exposes that ability. If it needs to change sizes, again, that can be exposed by a prop. But they aren't by default.

You can sort of accomplish this in "real" css using attributes carefully, but its not as elegant.