The self-assembly of two sizes of spherical nanocrystals has revealed a surprisingly diverse library of structures. To date, at least 15 distinct binary nanocrystal superlattice (BNSL) structures have been identified. The stability of these binary phases cannot be fully explained using the traditional conceptual framework treating the assembly process as entropy-driven crystallization of rigid spherical particles. Such deviation from hard sphere behavior may be explained by the soft and deformable layer of ligands that envelops the nanocrystals, which contributes significantly to the overall size and shape of assembling particles. In this work, we describe a set of experiments designed to elucidate the role of the ligand corona in shaping the thermodynamics and kinetics of BNSL assembly. Using hydrocarbon-capped Au and PbS nanocrystals as a model binary system, we systematically tuned the core radius (R) and ligand chain length (L) of particles and subsequently assembled them into binary superlattices. The resulting database of binary structures enabled a detailed analysis of the role of effective nanocrystal size ratio, as well as softness expressed as L/R, in directing the assembly of binary structures. This catalog of superlattices allowed us to not only study the frequency of different phases but to also systematically measure the geometric parameters of the BNSLs. This analysis allowed us to evaluate new theoretical models treating the cocrystallization of deformable spheres and to formulate new hypotheses about the factors affecting the nucleation and growth of the binary superlattices. Among other insights, our results suggest that the relative abundance of the binary phases observed may be explained not only by considerations of thermodynamic stability, but also by a postulated preordering of the binary fluid into local structures with icosahedral or polytetrahedral symmetry prior to nucleation.