Lactose intolerance occurs when the body doesn't produce enough lactase enzyme. Lactase is required for digesting lactose, a sugar found in milk and other dairy products. Lactose intolerance can be genetic or can be caused due to bacteria or viral infection, or any damage to the intestines.
You can learn more about lactose intolerance here.
Milk allergy is a kind of food allergy that involves the immune system. Here the body sees the proteins in dairy products as invaders and elicits an immune response that results in an allergic reaction. Milk/dairy allergy is one of the most common allergies in children.
You can learn more about milk allergy here.