Depletion effects on the structure and interactions between polymers induced by the presence of small depleting spheres are investigated by computer simulations. As the separation between two polymers decreases, the polymers repel each other due to the loss of conformational entropy. When the polymers are immersed in a medium crowded with small depleting spheres, however, depletion attractions between the polymer segments are induced. The resulting repulsive interaction is significantly reduced when the polymer segments approach one another closer than the size of the depleting spheres. The distance-dependent potential of mean force shows a highly nonmonotonic behavior reflecting the packing of the small depleting spheres around the polymer segments. We show that the depletion potential, that is, the component of the interactions arising from the presence of the small depleting spheres, between flexible polymers is qualitatively similar to that between two large spheres. However, there are small numerical differences that arise from the connectivity of the polymer chains. We also show that Brownian dynamics simulations of a single polymer chain with depletion potential can predict polymer statistical properties in good agreement with those from molecular dynamics simulations in which depleting spheres are explicitly accounted for. Therefore, we suggest the use of the depletion potentials for computational study of crowding effects on large biopolymers such as chromatin fibers.