Innovative nanofibrous scaffolds have attracted considerable attention in bone tissue engineering, due to their ability to mimic the hierarchical architecture of an extracellular matrix. Aiming at investigating how the polymer chemistry and fiber orientation of electrospun scaffolds (ES) based on poly(butylene succinate) (PBS) and poly(butylene succinate/diglycolate) (P(BS80BDG20)) affect human osteoblast differentiation, uniaxially aligned (a-) and randomly (r-) distributed nanofibers were produced. Although human osteoblastic SAOS-2 cells were shown to be viable and adherent onto all ES materials, a-P(BS80BDG20) exhibited the best performance both in terms of cellular phosphorylated focal adhesion kinase expression and in terms of alkaline phosphatase activity, calcified bone matrix deposition and quantitative gene expression of bone specific markers during differentiation. It has been hypothesized that the presence of ether linkages may lead to an increased density of hydrogen bond acceptors along the P(BS80BDG20) backbone, which, by interacting with cell membrane components, can in turn promote a better cell attachment on the copolymer mats with respect to the PBS homopolymer. Furthermore, although displaying the same chemical structure, r-P(BS80BDG20) scaffolds showed a reduced cell attachment and osteogenic differentiation in comparison with a-P(BS80BDG20), evidencing the importance of nanofiber alignment. Thus, the coupled action of polymer chemical structure and nanofiber alignment played a significant role in promoting the biological interaction.