To put it short and sweetly, casein is the protein and lactose is the sugar found in mammalian milk.
Casein composes about 80% of the protein found in animal milk, and provides essential amino acids, carbohydrates, calcium, and phosphorus. It has a chemical structure very similar to that of gluten, as gluten is also a protein (wheat, barley, rye). Like gluten, casein has binding properties, and is used in everything from paint (NOT for eating), protein powder, glue (only for eating by select children), and cheesemaking.
Cheesemaking is a good example of where the difference between casein and lactose can be clearly seen. Some aged cheeses are lactose-free, as the sugar is cultured out of the milk in the cheesemaking process. However, these cheeses still contain ample amount of the casein protein that will not age out, and serves as an integral part of the cheesiness of the cheese.
Lactose, on the other hand, only composes about 2-8% of milk. While casein has several uses, lactose’s only real use is by pharmaceutical companies as a filler, as it is relatively tasteless, cheap, and has a high compressibility. Lactose intolerance occurs when the human body does not supply adequate levels of lactase to digest the lactose sugars found in milk. The condition is most common in Asian, Native American, Mediterranean, and African ethnicities, as compared to those of Northern and Western Europe descent.
While a lactose intolerance may provide discomfort, since it is simply a lack of digestive enzyme, it is very, very seldom to be life-threatening. Casein, however, is a protein, and when people are allergic to proteins, results are an immediate-response-reaction. (a.k.a. could be fatal, and is not simply indigestion.)
Lesson learned: if someone says, “I’m allergic to milk,” what they are technically saying is that they are allergic to casein, the protein found in milk. If this is not the case, then “lactose intolerant” is a far more appropriate term, and a way more common condition in the American population.