Self-assembly of nanocrystals into functional materials requires precise control over nanoparticle interactions in solution that are dominated by organic ligands that densely cover the surface of nanocrystals. Recent experiments have demonstrated that small truncated-octahedral nanocrystals can self-assemble into a range of superstructures with different translational and orientational order of nanocrystals. The origin of this structural diversity remains unclear. Here, we use molecular dynamics computer simulations to study the self-assembly of these nanocrystals over a broad range of ligand lengths and solvent conditions. Our model, which is based on a coarse-grained description of ligands and solvent effects, reproduces the experimentally observed superstructures, including recently observed superlattices with partial and short-ranged orientational alignment of nanocrystals. We show that small differences in nanoparticle shape, ligand length and coverage, and solvent conditions can lead to markedly different self-assembled superstructures due to subtle changes in the free energetics of ligand interactions. Our results rationalize the large variety of different reported superlattices self-assembled from seemingly similar particles and can serve as a guide for the targeted self-assembly of nanocrystal superstructures.