We study the main astrophysical properties of differentially rotating neutron stars described as stationary and axisymmetric configurations of a moderately stiff ${\rm{\Gamma }}=2$ polytropic fluid. The high level of accuracy and of stability of our relativistic multidomain pseudo-spectral code enables us to explore the whole solution space for broad ranges of the degree of differential rotation, but also of the stellar density and oblateness. Staying within an astrophysically motivated range of rotation profiles, we investigate the characteristics of neutron stars with maximal mass for all types of families of differentially rotating relativistic objects identified in a previous article. We find that the maximum mass depends on both the degree of differential rotation and the type of solution. It turns out that the maximum allowed mass can be up to 4 times higher than what it is for nonrotating stars with the same equation of state. Such values are obtained for a modest degree of differential rotation but for one of the newly discovered types of solutions. Since such configurations of stars are not that extreme, this result may have important consequences for the gravitational wave signal expected from coalescing neutron star binaries or from some supernova events.