In the framework of non-isochronous variations, the Pfaff-Birkhoff variational principle is first restructured and then the Birkhoffian equations and the Pfaffian 1-form are rederived from the reformed principle correspondingly. The classical Noether theorem for Birkhoffian systems is further reformulated in a geometric way, which reveals the relationship between the invariance of the Pfaffian 1-form under Lie group actions and associated conservation laws. In parallel with the continuous case, the discrete analogues of the Pfaff-Birkhoff variational principle, the Birkhoffian equations and the Pfaffian 1-form are constructed successively from the discretized Pfaffian action sum, which is an approximation of the Pfaffian action integral with adaptive time steps. For discrete Birkhoffian systems, i.e., systems characterized by the discrete Birkhoffian equations particularly, the invariance of the discrete Pfaffian 1-form under Lie group actions also results in a conserved discrete momentum map, defined by the contraction of the corresponding infinitesimal generator and the discrete Pfaffian 1-form.