Milk allergy and lactose intolerance are two entirely different issues.
A person with a milk allergy may have a dangerous or unpleasant allergic (immune) reaction to one of the components of milk (from any animal). The substance most likely to cause this reaction is the milk protein called casein. The response can range from a simple unpleasant feeling to hives and skin rashes, or to a full anaphylactic reaction, which can be life-threatening. Milk protein intolerance is a related condition.
Lactose intolerance is not an allergy since no autoimmune response is involved. It occurs in people who have difficulty digesting the milk sugar called lactose. Bloating, gas, diarrhea, and other intestinal symptoms are often the result. It is an inherited condition, but one often not diagnosed until later in life.