I think nylon can be as stiff, whether stiffness matters there isn't much good data on (likely because the gains are very, very small).
However, the carbon shoes, for the same stiffness and features, will be lighter, sometimes considerably so. I think that's the main benefit. I have some Shimano SPD-SL R315, which were top of the line way back in the old days of 2011 when I got them and are carbon. They're pretty light, at 530g for a pair of 40s, where even the newest Specialized S-Works stuff is 540g for a pair of 42s.
Compared to the nylon Shimano shoes they replaced, they feel considerably lighter in your hand.