Rock crystal: the "transparent" quartz
Rock crystal is basically quartz in its purest, cleanest form. When it formed, conditions were so stable and the crystals grew so neatly that the result is transparent, almost like natural glass, but much harder.
In fact, for centuries people mistook rock crystal for eternal ice that never melted, which is part of where its name comes from in other languages (in Spanish it's called "cristal de roca", literally "rock crystal"). If you hold it in your hand, you'll notice it feels cold to the touch, surprisingly heavy for its size, and that light passes through it almost effortlessly.
In short: rock crystal = transparent quartz, with barely any impurities, and a highly ordered crystal structure.
White quartz: rock crystal's "milky" cousin
Here's the part that confuses people the most. White quartz (also called milky quartz) is chemically the same as rock crystal: silica. The difference lies in how it formed.
As it grew, white quartz trapped countless microscopic bubbles of air and water inside its structure. They're far too small to see with the naked eye, but there are so many of them that, together, they stop light from passing straight through the stone and scatter it in every direction instead. The result? Instead of seeing through it like rock crystal, you see an opaque white, like milk or compacted snow.
So when you hold a piece of white quartz in one hand and rock crystal in the other, you're really holding the same stone, but with different "formation stories". One grew calmly and in an orderly way (rock crystal); the other grew faster or under more turbulent conditions, trapping air and water along the way (white quartz).
And here's the detail that really makes the difference: in rock crystal, the silicon and oxygen atoms had the time and stable conditions needed to line up one after another in a perfectly ordered pattern, forming that characteristic crystal structure of quartz. In white quartz, on the other hand, that order was interrupted: the trapped air and water bubbles broke up the continuity of that ordered network, so although it's still quartz, its internal structure is far more irregular. That's why one lets light pass through cleanly while the other scatters it in all directions.