Before revision:  We also prove that every spectral sparsifier of the clique having average degree d and a certain high "pseudo-girth" property has an approximation error that is at least the "Ramanujan bound" (2-o_{n,d}(1))/\sqrt d, which is met by d-regular Ramanujan graphs, generalizing a lower bound of Srivastava and Trevisan \mbox{%DIFAUXCMD ST18 Together, these results imply a separation between spectral sparsification and cut sparsification.
After revision:  We also prove that every spectral sparsifier of the clique having average degree d and a certain high "pseudo-girth" property has an approximation error that is at least the "Ramanujan bound" (2-o_{n,d}(1))/\sqrt d, which is met by d-regular Ramanujan graphs, generalizing a lower bound of Srivastava-Trevisan '18. Together, these results imply a separation between spectral sparsification and cut sparsification.