Concrete material model plays an important role in numerical predictions of its dynamic responses subjected to projectile impact and charge explosion. Current concrete material models could be distinguished into two kinds, i.e., the hydro-elastoplastic-damage model with independent equation of state and the cap-elastoplastic-damage model with continuous cap surface. The essential differences between the two kind models are vital for researchers to choose an appropriate kind of concrete material model for their concerned problems, while existing studies have contradictory conclusions. To resolve this issue,the constitutive theories of the two kinds of models are firstly overviewed. Then, the constitutive theories between the two kinds of models are comprehensively compared and the main similarities and differences are clarified, which are demonstrated by single element numerical examples. Finally, numerical predictions for projectile penetration and charge explosion experiments on concrete targets are compared to further demonstrate the conclusion made by constitutive comparison. It is found that both the two kind models could be used to simulate the dynamic responses of concrete under projectile impact and blast loadings, if the parameter needed in material models are well calibrated, although some discrepancies between them may exist.