Differential cross sections have been extracted from exclusive and kinematically complete high-statistics measurements of quasifree polarized $\vec{n}p$ scattering performed in the energy region of the $d^*(2380)$ dibaryon resonance covering the the range of beam energies $T_n$ = 0.98 - 1.29 GeV ($\sqrt s$ = 2.32 - 2.44 GeV). The experiment was carried out with the WASA-at-COSY setup having a polarized deuteron beam impinged on the hydrogen pellet target and utilizing the quasifree process $dp \to np + p_{spectator}$. That way the $np$ differential cross section $��(��)$ was measured over a large angular range. The obtained angular distributions complement the corresponding analyzing power $A_y(��)$ measurements published previously. A SAID partial-wave analysis incorporating the new data strengthens the finding of a resonance pole in the coupled $^3D_3 - ^3G_3$ waves.