As far as I know, there is no big difference between celiac and coeliac disease. Both words refer to a particular disease that is developed in the body. The major difference between the two is that both terms are not spelled in the same way, i.e., one contains letter O, whereas the letter is missing in the second word.
The one that contains letter O is usually considered as the right word in most European countries. Coeliac or celiac is also known as gluten enteropathy, tropical sprue, etc. It is a particular disease that is developed in the intestine. The villi of the intestine are usually affected when they react to certain food materials that contain Gluten.
Gluten usually represents the major protein in cereals such as rye, wheat, barley, etc. Once gluten is detected in a food, the immune system responds by seeing gluten is a poison, and in the process, the villi of the intestine are destroyed.