We experimentally investigate two-photon interference between a weak coherent state and a heralded single-photon state, producing from a spontaneous parametric down conversion. Both the unequal spectrum linewidth and average photon number ratio in a given time interval are considered in our model. We obtained excellent agreement between our experimental data and prediction from our model. Furthermore, the range of observing high visibility two-photon interference is significantly extended by isolating coincidence events from two-photon contributions in the weak coherent state. These results may throw some new light on quantum information technology when the two-photon interference with independent sources is required.