We used non-equilibrium molecular dynamics (Langevin NEMD) alongside equilibrium molecular dynamics (EMD) simulations to investigate the Soret coefficient of binary Lennard-Jones mixtures. The Langevin NEMD simulations served as benchmarks because EMD predictions based on the Green-Kubo formalism are challenging due to a weak signal-to-noise ratio of the cross-correlation functions, the strong system-size dependence of the mass-mass phenomenological coefficient and the uncertainties related to the calculation of the thermodynamic factor. Once optimal settings for the EMD simulations were determined, the dataset from the EMD simulations was successfully used to rationalize the behavior of the Soret coefficient. Furthermore, the EMD data were leveraged to test the applicability of a lean theoretical model, which allows for predictions of the Soret coefficient mainly based on thermodynamic data.