The computational efficiency and the stability of Continuous Galerkin (CG) methods, with Taylor–Hood approximations, and Hybridizable Discontinuous Galerkin (HDG) methods are compared for the solution of the incompressible Stokes and Navier–Stokes equations at low Reynolds numbers using direct solvers. A thorough comparison in terms of CPU time and accuracy for both discretization methods is made, under the same platform, for steady state problems, with triangular and quadrilateral elements of degree $k=2-9$. Various results are presented such as error vs. CPU time of the direct solver, error vs. ratio of CPU times of HDG to CG, etc. CG can outperform HDG when the CPU time, for a given degree and mesh, is considered. However, for high degree of approximation, HDG is computationally more efficient than CG, for a given level of accuracy, as HDG produces lesser error than CG for a given mesh and degree. Finally, stability of HDG and CG is studied using a manufactured solution that produces a sharp boundary layer, confirming that HDG provides smooth converged solutions for Reynolds numbers higher than CG, in the presence of sharp fronts.