Spatiotemporal correlations of the two-dimensional spring-block (Burridge-Knopoff) model of earthquakes are extensively studied by means of numerical computer simulations. The model is found to exhibit either a “subcritical”, “supercritical” or “near-critical” behavior, depending on the values of the model parameters. Transition between the “subcritical” and “supercritical” regimes is either continuous or discontinuous. Seismic events in the “subcritical” regime and those in the “supercritical” regime at larger magnitudes exhibit universal scaling properties. In the “supercritical” regime, eminent spatiotemporal correlations, e.g., a remarkable growth of seismic activity preceding the main shock, arise in earthquake occurrence, whereas such spatiotemporal correlations are significantly suppressed in the “subcritical” regime. Seismic activity is generically suppressed just before the main shock in a close vicinity of the epicenter of the upcoming event while it remains to be active in the surroundings (the Mogi doughnut). It is also observed that, before and after the main shock, an apparent B-value of the magnitude distribution decreases or increases in the “supercritical” or “subcritical” regimes, respectively. Such distinct precursory phenomena may open a way to the prediction of the upcoming large event.