Carbon in a wheel increases the strength and stiffness of the rim, which in turn helps reduce the loss of power produced by the rider, according to "High-Performance Composites" by Jared Nelson. Aluminum is less stiff and more likely to wobble and vibrate because it cannot maintain the shape of the wheel as well as carbon.
Less carbon than aluminum is needed to make a stiff rim, resulting in lighter wheels and significantly increased rider comfort. This is especially important in performance or racing bicycles.
The manufacture of tubular carbon rims was previously much more labor-intensive and required more steps than aluminum wheels. When manufacturers started making the tubes separately, as is true for consumer-level bicycles, manufacturing costs fell.
