Both are low temperature injuries, but for freeze damage to occur, the product must be below its freezing point.  All products contain some sugar which means that the freezing point is always lower than that of water (32°F; 0°C). The exact temperature below 32°F at which the product will freeze will depend on the amount of sugar or soluble carbohydrate. For example, lettuce with a low sugar content may freeze at 31.7°F (-0.2°C) while plums with a very high sugar content may freeze at 29°F (-1.7°C) or lower.  Much of the damage due to freezing results from the formation of ice crystals and the destruction of cell integrity once the product thaws, leading to the frequent symptoms of water soaked appearance, mushy texture, and complete collapse and breakdown of the damaged areas.   

Chilling injury occurs at a range of temperatures that are low but nonfreezing for that product.  The development of chilling injury depends on the specific temperature and time of exposure (see basil in the Herbs Produce facts at http://postharvest.ucdavis.edu/pfvegetable/Herbs/ for a good example). Products differ in their susceptibility to chilling injury (see differences among eggplant types at http://postharvest.ucdavis.edu/pfvegetable/Eggplant/). Bananas are extremely chilling sensitive if stored below 55°F (12.5°C) for a few days while honeydew melons require weeks to show chilling symptoms at 41°F (5°C).  Chilling injury symptoms vary depending on the commodity but often include surface discoloration, surface pitting, water soaked areas, lack of ability to ripen in fruits, and increased decay, among others.  Another consideration for chilling injury is that sometimes the symptoms do not appear or are not as severe until the product has been removed from the low temperature storage and warmed (as can occur during marketing). 

-Marita Cantwell