Geometric complexity theory is an approach towards the separation of fundamental algebraic complexity classes. Two papers by Mulmuley and Sohoni [K. D. Mulmuley and M. Sohoni, SIAM J. Comput., 31 (2001), pp. 496--526; SIAM J. Comput., 38 (2008), pp. 1175--1206] pursue this goal via representation theoretic multiplicities in coordinate rings of specific group varieties. The papers also conjecture that the vanishing behavior of these multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions was recently disproved in 2016 in two successive papers, [C. Ikenmeyer and G. Panova, Adv. Math., 319 (2017), pp. 40--66] and [P. Bürgisser, C. Ikenmeyer, and G. Panova, J. Amer. Math. Soc., 32 (2019), pp. 163--193]. This raises the question of whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. This paper provides for the first time a setting where separating with multiplicities can be achieved, while separation with occurrences is provably impossible. Our setting is surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e., a higher secant variety of the Veronese variety). As a side result we prove a slight generalization of Hermite's reciprocity theorem, which proves Foulkes's conjecture for a new infinite family of cases.