We conducted simultaneous, small-angle, X-ray scattering/differential scanning calorimetry (SAXS/DSC) and simultaneous, wide-angle, X-ray scattering (WAXS)/DSC measurements for a polymer blend of poly($varepsilon$-caprolactone)/poly(ethylene glycol)(PCL/PEG). The time-dependent SAXS/DSC and WAXS/DSC results, measured while the system was quenched below the melting temperature of PCL from a melting state, revealed the competitive behavior between liquid-liquid phase separation and crystallization in the polymer blend. The time-dependent structural evolution extracted from the SAXS/WAXS/DSC results can be characterized by the following four stages in the PCL crystallization process: the induction (I), nucleation (II), growth (III), and late (IV) stages. The influence of the liquid-liquid phase separation on the crystallization of PCL was also observed by phase-contrast microscope and polarized microscope with 1/4$lambda$ compensator.