As a disclaimer, I am not a doctor. I am not an expert. I am, however, an individual who suffers from both intolerances and food allergies.
A dairy allergy is when an body’s immune system mistakenly recognizes dairy protein as dangerous and attacks it. There are two proteins in milk, both can cause an allergic reaction and you can be allergic to one or both proteins: casein and whey. Allergic reactions can be as benign as hives (however annoying) and as dangerous as anaphylaxis. Allergies can get worse over time and the more you react to that allergen (for example, apples used to make my throat itchy and now my husband cannot kiss me if he has had an apple recently due to my throat closing). My non-professional opinion when dealing with any food allergy is to see a professional specializing in allergies.
Lactose intolerance relates to the body’s inability to digest the sugar lactose due to a deficiency in the lactase enzyme. Humans produce lactase until about the age of five and are the only animal to drink milk past the infancy stage. As we age, we naturally become lactase deficient and thus lactose intolerant. The degrees of lactose intolerance vary, with some people able to eat certain dairy products or all dairy products in small amounts and other people unable to eat any product which contains lactose. Lactase pills and milk are also available, however they work better for some than others.