A diamond has more refractive index (2.4>1.5) and less critical angle(24°<42° ) as compared to  glass.

A diamond is cut in a special way such that many total internal reflections take place inside it. After multiple such reflections, the colors in the light are separated and seen individually. This phenomenon causes the diamond to shine in dim light as well.

However, such is not possible for glass due to its inability to bend light as compared to diamond. Hence, a diamond appears to shine but a piece of glass cut to the same shape does not shine.