An efficient finite-rate chemistry (FRC) - large-eddy simulation (LES) formulation is developed for numerical modeling of complicated turbulent combustion, using a point-implicit stiff ODE solver and a correlated dynamic adaptive chemistry algorithm. Compared to conventional brute force FRC-LES approach, the new FRC-LES approach provides a speed up of 8.6 times for chemistry, and reduces the total computation time by 6.4 times. With this new framework, simulations using both the FRC-LES and the flamelet/progress-variable (FPV)-LES approaches are conducted for a piloted partially premixed methane/air flame, which contains low-level of local extinction and re-ignition. The axial and radial profiles of time-averaged statistics, including temperature, mixture fraction, and major species mass fractions, using these two approaches show good agreement with the experiment data. In the downstream region, both models predict more partially premixed burning instead of stoichiometric burning, such that their prediction of fuel-to-product conversion and heat release are lower than experimental data, especially on the fuel rich side. Although both cases show very similar time-averaged flame fields, instantaneously, the FPV-LES approach predicts significantly larger level of local extinction than the FRC-LES case, especially in the downstream region. Near the stoichiometric region, the axial distributions and conditional statistics indicate that the FPV-LES approach over-predicts the fuel consumption, but under-predicts the radical-to-product conversion and heat release, which explains its over-prediction of local extinction level. In contrast, on the fuel rich side, the conditional statistics show that the FPV-LES approach under-predicts the level of local extinction, due to its over-prediction of fuel-to-product conversion and heat release there. The conditional statistics indicates that the FRC-LES approach provides better predictions for temperature and most species at different axial locations. In summary, the new FRC-LES formulation developed in this work not only provides a better predication than the FPV-LES method, but is also significantly more efficient than the conventional FRC-LES method.