Similarity: They provide the uniqueness of the density associated with the external potential and its Hamiltonian showing the one-to-one correspondence. The functional is of the ground state charge density, where the Hamiltonian is a functional of the ground state charge density.

Difference: They provide a mathematical relation between to potentials if they have the same ground-state energy then the difference between them is constant. HK functional does not explicitly depend on the external potential.