Question

why does a diamond shine more than a glass piece cut to the same shape

Answer 1

The amount of "sparkle" an optical material provides (assuming the same cut as stated in the question details and also the same quality of polish) is primarily determined by its dispersion: how much the index of refraction varies as a function of wavelength in the visible range (390-700nm).  Additional contributions come from the overall larger index of refraction.

Answer 2

The critical angle of a diamond is very low (i.e.,24.4°). So if a light ray enter a diamond it is definitely undergo total internal reflection. Where as it is not possible with glass pieces cut to the same shape. So diamond shine more than a glass pieces.