A detailed comparative numerical study between the two-dimensional (2D) and quasi-two-dimensional (quasi-2D) turbulent Rayleigh-Bénard (RB) convection on flow state, heat transfer, and thermal dissipation rate (TDR) is made. The Rayleigh number (Ra) in our simulations ranges up to 5×10^{10} and Prandtl number (Pr) is fixed to be 0.7. Our simulations are conducted on the Tianhe-2 supercomputer. We use an in-house code with high parallelization efficiency, based on the extended PDM-DNS scheme. The comparison shows that after a certain Ra, plumes with round shape, which is called the “temperature islands”, develop and gradually dominate the flow field in the 2D case. On the other hand, in quasi-2D cases, plumes remain mushroom-like. This difference in morphology becomes more significant as Ra increases, as with the motion of plumes near the top and bottom plates. The exponents of the power-law relation between the Nusselt number (Nu) and Ra are 0.3 for both two cases, and the fitting pre-factors are 0.099 and 0.133 for 2D and quasi-2D respectively, indicating a clear difference in magnitude of the heat transfer rate between two cases. To understand this difference in the magnitude of Nu, we compare the vertical profile of the horizontally averaged TDR for both two cases. It is found that the profiles of both cases are nearly the same in the bulk, but they vary near boundaries. Comparing the bifurcation height z_{b} with the thermal boundary layer thickness δ_{θ}, it shows that z_{b} < δ_{θ}(3D) < δ_{θ}(2D) and all three heights obey a universal power-law relation z~Ra^{-0.30}. In order to quantify the difference further, we separate the domain by z_{b}, i.e., define the area between two z_{b} (near top and bottom plates respectively) as the “mid region” and the rest as the “side region”, and integrate TDR in corresponding regions. By comparing the integral it is found that most of the difference in TDR between two cases, which is connected to the heat transfer rate, occurs within the thermal boundary layers. We also compare the ratio of contributions to total heat transfer in BL-bulk separation and side-mid separation.