  Let $X$ be a cubic fourfold that has only simple singularities and does not
contain a plane. We prove that the Fano variety of lines on $X$ has the same
analytic type of singularity as the Hilbert scheme of two points on a surface
with only ADE-singularities. This is shown as a corollary to the
characterization of a singularity that is obtained as a $K3^{[2]}$-type
contraction and has a unique symplectic resolution.
