We show that the increase of the interface free energy with deviation from equilibrium seen in recent Monte Carlo simulations [S. Auer and D. Frenkel, Nature (London) 413, 711 (2001)] can be recovered if the molecular scale diffuseness of the crystal-liquid interface is considered. We compare two models, Granasy's phenomenological diffuse interface theory, and a density functional theory that relies on the type of Ginzburg-Landau expansion for fcc nucleation, that Shih introduced for bcc crystal. It is shown that, in the range of Monte Carlo simulations, the nucleation rate of the stable fcc phase is by several orders of magnitude higher than for the metastable bcc phase, seen to nucleate first in other fcc systems. The nucleation barrier that the diffuse interface theories predict for small deviations from equilibrium is in far better agreement with the simulations than the classical droplet model. The behavior expected at high densities is model dependent. Granasy's phenomenological diffuse interface theory indicates a spinodal point close to glass transition, while a nonsingular behavior is predicted by the density functional theory with constant Ginzburg-Landau coefficients. Remarkably, a minimum of the nucleation barrier, similar to the one seen in polydisperse systems, occurs if the known density dependence of the Ginzburg-Landau coefficients is considered.