Is lactose intolerance and a milk allergy the same thing?

Different: The bodily response responsible for intolerance and allergy are different, although they are frequently confused.

No: Lactose intolerance is the inability to break down lactose due to endogenous deficiency of lactase, an enzyme that breaks down lactose into simpler sugars. On the other hand milk allergy is an IgE mediated antibody response which leads to histamine release from cells and the potential systemic symptoms of an allergic reaction.