Asymmetric patchy particle models have recently been shown to describe the crystallization of small globular proteins with near-quantitative accuracy. Here, we investigate how asymmetry in patch geometry and bond energy generally impacts the phase diagram and nucleation dynamics of this family of soft matter models. We find the role of the geometry asymmetry to be weak, but the energy asymmetry to markedly interfere with the crystallization thermodynamics and kinetics. These results provide a rationale for the success and occasional failure of the proposal of George and Wilson for protein crystallization conditions as well as physical guidance for developing more effective protein crystallization strategies.