The main difference between quartz crystal and glass is that glass does not have a crystalline structure. Glass is also made with additives that quartz crystal does not contain.
Pure quartz crystal consists of nothing but a lattice of silicon and oxygen. Each silicon atom is bonded to four oxygen atoms, and each oxygen atom is bonded to two silicon atoms, making it silicon dioxide. Glass largely consists of silicon dioxide as well, but the molecules in glass are not arranged in a crystalline structure. Most commercial glass is only 75 percent silicon dioxide; the rest consists of additives like sodium oxide and calcium oxide in the form of lime. Lead is also a common additive to certain types of glass.
Visible light is nearly completely permeable to clear glass, and a flat pane of glass does not affect its properties. Quartz crystal is permeable to visible light but polarizes it. The faces of the crystal through which it passes affect this polarization.
Quartz crystal also has a useful application, called piezoelectricity, that glass does not. Piezoelectricity is the generation of electrical potential through mechanical stresses, and the opposite is also true. This relationship is used to vibrate quartz to keep precise time in watches and computer clocks.