What is the difference between fractional freezing and fractional crystallization? More simply put, what is the difference between freezing and precipitation when considering a mixture?

Take two scenarios:

First a mineral in water. If the temperature of this solution is lowered, the solubility of the mineral will decrease and eventually precipitate out of solution. This could be done in a cascade type setup so that each time, the solution is cooled further and the liquid phase becomes more and more concentrated in water.

Second, consider a mixture of ethanol and water. Here too, I could lower the temperature until crystals first form. They would be almost pure water, leaving behind a solution higher in concentration in alcohol. I could repeat this procedure until I reach the eutectic mixture. 

In both cases, a drop in temperature is allowing a solid phase to form out of an aqueous solution, yet one is referred to as partial crystallization and the other is referred to as partial freezing. I am not sure I see the difference.
There is no difference. Strictly speaking, "freezing" does not necessarily include crystalline solids, and may include amorphous materials that precipitate out of solution. But, this does not really matter in the context of these eutectic (or similar) systems.

I suspect the latter is referred to as "partial freezing" because the solid is water, whereas it is more general in the former. We rarely speak of water crystallising to ice, but rather water freezing to ice. This is more of a linguistic issue than scientific.