Carbon Tri/Cycling Shoe?

The biggest advantage of CF sole shoes is that you can vary the layup to make it very stiff in one direction yet very compliant in another. To a smaller degree you can also control stiffness in certain locations to reduce the "hot spot symptom".

And yes, they tend to be lighter. Most important with tri shoes though is the fit, even more so than road shoes. Make sure you try them out before buying your first pair. Once you know which ones work well for you, buying replacement shoes online is easy.

Thickness can also be effected by the type of material. Generally CF allows the sole to be thinner (at a given stiffness) which lowers the stack height over the pedal. Lower stack height means more direct power transfer and also a less likelihood of hot spots.

Typically its not worth the extra cost.

I have carbon Specialized TriVents for Tri, standard LG road cycling shoes and carbon LG MTB shoes.
The only reason I got carbon on my Tri and MTB shoes is becasue they were on sale. I don't like the MTB shoes 'coz they don't fit great.