•Entropy and entransy theories are reviewed and compared from three perspectives.•Multi-objective optimization with entropy generation cannot meet various demands.•Minimum entropy generation does not lead to maximum heat transfer coefficient.•Entransy dissipation extremum corresponds to maximum heat transfer coefficient.•Entransy with Pareto Optimality is more appropriate for heat transfer optimization. Performance improvement of convective heat transfer processes is significant for energy conservation. Considering the trade-off between the heat transfer enhancement and the pumping power reduction, the optimization of convective heat transfer processes can be modeled as a multi-objective optimization problem in the entropy-based approach and a constrained optimization problem in the entransy-based approach. This article first reviews these two theories and then compares them from the perspectives of optimization criteria, optimization methods and optimization results. Studies have shown that simply analyzing the entropy generation rate or other entropy generation criteria cannot meet the diverse objectives of various practical applications. Besides, the minimum heat transfer entropy generation does not always lead to the maximum of convective heat transfer coefficient, as reason is also analyzed here. In contrast, the entransy dissipation extremum corresponds to the maximum convective heat transfer coefficient as has been shown mathematically. Moreover, the entransy dissipation extremum principle and the variational method can be combined to find the optimal flow and temperature fields with better heat transfer results than determined using the entropy generation minimization principle. The entransy optimization results can then be used to design better augmentation technologies for convective heat transfer processes. In summary, the entransy theory is more appropriate for optimizing convective heat transfer processes without heat-work conversion processes.