Previously, a new biodegradable poly(ester urethane urea) was synthesized based on polycaprolactone-diol and fish gelatin (PU-Gel). In this work, the potential of this new material for neural tissue engineering is evaluated. Membranes with randomly oriented fibers and with aligned fibers are produced using electrospinning and characterized regarding their mechanical behavior under both dry and wet conditions. Wet samples exhibit a lower Young's modulus than dry ones and aligned membranes are stiffer and more brittle than those randomly oriented. Cyclic tensile tests are conducted and high values for recovery ratio and resilience are obtained. Both membranes exhibited a hydrophobic surface, measured by the water contact angle (WCA). Human mesenchymal stem cells from umbilical cord tissue (UC-MSCs) and human neural stem cells (NSCs) are seeded on both types of membranes, which support their adhesion and proliferation. Cells stained for the cytoskeleton and nucleus in membranes with aligned fibers display an elongated morphology following the alignment direction. As the culture time increased, higher cell viability is obtained on randomfibers for UC-MSCs while no differences are observed for NSCs. The membranes support neuronal differentiation of NSCs, as evidenced by markers for a neuronal filament protein (NF70) and for a microtubule-associated protein (MAP2).