  We revisit the work of Lehn-Lehn-Sorger-van Straten on twisted cubic curves
in a cubic fourfold not containing a plane in terms of moduli spaces. We show
that the blow-up $Z'$ along the cubic of the irreducible holomorphic symplectic
eightfold $Z$, described by the four authors, is isomorphic to an irreducible
component of a moduli space of Gieseker stable torsion sheaves or rank three
torsion free sheaves. For a very general such cubic fourfold, we show that $Z$
is isomorphic to a connected component of a moduli space of tilt-stable objects
in the derived category and to a moduli space of Bridgeland stable objects in
the Kuznetsov component. Moreover, the contraction between $Z'$ and $Z$ is
realized as a wall-crossing in tilt-stability. Finally, $Z$ is birational to an
irreducible component of Gieseker stable aCM bundles of rank six.
