The differential cross sections of radiative-collisional quenching (absorption) of a metastable state of a colliding atom are calculated for the first time. As a particular example, the reaction of quenching (excitation) of the metastable state He(2¹S) in collisions with Ne atoms in the ground state is considered. The calculations are performed for the thermal collisional energy E = 10^-3 au for a wide range of radiation frequencies, including both wings and the center of the line of a forbidden atomic transition, and are based on the uniform quasi-classical approximation, which generalizes the Franck-Condon approximation to the case of an exponential dependence of the transition dipole moment on the internuclear distance, as well as to the case of nonintersecting terms. The calculated differential cross sections have an oscillating structure, which, for the blue wing of the forbidden spectral line, is interpreted as Stückelberg oscillations. At a radiation frequency close to the frequency of the forbidden atomic transition, a sharp maximum in the differential cross section - the giant glory effect - is observed in the range of small scattering angles. This effect is shown to occur as a result of superimposition of the rainbowlike feature of the differential cross section on the glory feature.