For me, there is a clear difference. The 1st says that there is a functional that gives you the energy of the ground state. I see it as a one way implication density => GS energy; however, the second theorem means that if you have this GS energy functional, you can use it to find the density by searching for the density that minimizes it. So I look at it as GS energy functional => density. So when you put the two theorems together, it means that if you have the exact energy functional, you can use it to determine the density (and thus all other properties of a material). Or if you have the density, you can use it to determine the energy functional and all the other properties of a material.