Image from http://commons.wikimedia.org/wiki/File:Milk_jugs_in_a_row.jpg / Author: ChildofMidnight
Before I really get into it, let me just say this: Milk allergy and Lactose Intolerance are completely different things. While they both deal with dairy products, they are not the same thing. (And if you use milk allergy and lactose intolerance interchangeably after reading this, someone should hit you with a gallon of milk).
Okay, now that I've gotten that off my chest, let's begin. Lactose intolerance means that your body doesn't produce the enzyme, lactase, to break down the sugar, lactose, that is found in cows milk. Because the body isn't breaking down the sugar, people with lactose intolerance usually have stomach discomfort and a few other symptoms, but it generally doesn't last too long. There are also varying degrees of lactose intolerance. Some people's bodies do produce lactase, and can therefore break down some lactose, they just need to be careful to not have too much milk. This is pretty simplified, so if you want more details about lactose intolerance and how it differs from milk allergy, you can click here.
Now on to milk allergy, also known as dairy allergy. A milk allergy really isn't that much different than other food allergies. It's just weird. When someone has a milk allergy, it means that they are allergic to some protein that is found in cows milk, usually whey, casein, or both. Just like some people are allergic to cats, pollen, peanuts, and soaps, people can be allergic to whey and casein. Basically, the body treats the proteins as they would an infection, and the immune system attacks the proteins. This can cause a ton of different "allergic" reactions. Some of which may surprise you.
Obviously most people will have the abdominal pain/discomfort and maybe some vomiting and/or diarrhea (duh), but there are many other reactions that people can experience as well. Some of my typical reactions include a sore throat and runny nose (just like someone who is allergic to a cat), hives or a skin rash, and a migraine. For me, my reactions depend on the amount of dairy that I have ingested, and I almost always have a few different reactions.
Keep in mind that it might take a little while for you to discover your specific reactions. I knew that a skin rash was one of my reactions, before I even knew I had a milk allergy. It wasn't until after I gave up dairy (then accidentally ate something containing cows milk), that I realized that I also get the sore throat, runny nose, and migraines (if I have a large amount).
This is just my experience with milk allergy and what I have learned so far, but hopefully I have cleared up some confusion. If you have a milk allergy (or any food allergy) then I feel your pain. Good luck!