Caudiciforms separate jobs between their caudex and their branches and leaves. The caudex stores the water and the leaves conduct the process of photosynthesis.
A good way to understand this difference is by comparing caudiciform succulents to leafy succulents. Leafy succulents, like kalanchoes and echeverias, share the same food-processing and water-storing jobs throughout their entire anatomy. The green of their leaves process the food and with the help of mucilaginous cells they store water. If a leaf falls off a leafy succulent it doesn't wither right away. More than not it will grow roots and sprout a new plant!
The thin leaves of a caudiciform succulent have more stomata, or breathing pores, than leafy succulents and therefore lose water quickly. When a leaf falls, it withers and dies within a day or two without ever forming roots. This characteristic is mesophytic. When branches and leaves die from dehydration and extreme heat, the underground caudex survives.