We recently proposed a dynamical mesoscopic model for DNA, which is based, like the statistical ones, on site-dependent finite stacking and pairing enthalpies. In the present paper, we first describe how the parameters of this model are varied to get predictions in better agreement with experimental results that were not addressed up to now, like mechanical unzipping, the evolution of the critical temperature with sequence length and temperature resolution. We show that the model with the new parameters provides results that are in quantitative agreement with those obtained from statistical models. Investigation of the critical properties of the dynamical model suggests that DNA denaturation looks like a first-order phase transition in a broad temperature interval, but that there necessarily exists, very close to the critical temperature, a crossover to another regime. The exact nature of the melting dynamics in this second regime still has to be elucidated. We finally point out that the descriptions of the physics of the melting transition inferred from statistical and dynamical models are not completely identical and discuss the relevance of our model from the biological point of view.