Hi!
The main difference between an allergy and an intolerance is that an allergy is an immune response, while an intolerance is a reaction of the digestive system. With an allergy, the immune system mistakes a harmless food protein as dangerous and mounts an immune response against it. With an intolerance, the food either irritates the digestive tract or the person is unable to digest it (like lactose intolerance). Here's a website with a really good explanation and symptoms, etc:
http://www.webmd.com/allergies/foods-allergy-intoleranceUnfortunately, nausea and diarrhea are symptoms of both intolerance and allergy. To know for sure you may have to get a blood test done
Given the severity of her reaction I would keep her off all dairy myself, at least until you can discuss it with your doctor.
Hth
Vikki