  Motivated by the question of rationality of cubic fourfolds, we show that a
cubic X has an associated K3 surface in the sense of Hassett if and only if the
variety F of lines on X is birational to a moduli space of sheaves on a K3
surface, but that having F birational to Hilb^2(K3) is more restrictive. We
compare the loci in the moduli space of cubics where each condition is
satisfied.
