Using fast lattice Monte Carlo (FLMC) simulations both in a canonical ensemble and with Wang-Landau-transition-matrix sampling, we have studied a model system of homopolymer brushes in an implicit, good solvent. Direct comparisons of the simulation results with those from the corresponding lattice self-consistent field (LSCF) theory, both of which are based on the same Hamiltonian (thus without any parameter-fitting between them), unambiguously and quantitatively reveal the fluctuations and correlations in the system. We have examined in detail how the chain number density C and the interaction strength N/kappa (where N is the number of segments on a chain and kappa is inversely proportional to the second virial coefficient characterizing the solvent quality) affect both the brush structures and thermodynamics. For our model system, the LSCF theory is exact in both limits of C -> infinity and N/kappa -> infinity, where there are no fluctuations or correlationS. At finite C and N/kappa > 0, the segmental density profile in the direction perpendicular to the grafting substrate obtained from FLMC simulations is flatter than the LSCF prediction, and the profile differences are smaller in higher dimensions. The free end density from FLMC simulations is also lower than the LSCF prediction well inside the brush. The LSCF theory underestimates the free energy but overestimates the entropy per chain, and underestimates the internal energy per chain at small N/kappa but overestimates it at large N/kappa. At large C, FLMC results approach LSCF predictions at a rate of 1/C in most cases.