If you have Celiac Disease you have a gluten intolerance, but if you have a gluten intolerance it doesn't mean you have Celiac Disease.
Celiac Disease is an autoimmune disease (like Lupus, Hashimotos and Juvenile Diabetes). It basically means that your immune system confuses itself for a foreign body and attacks itself. Celiac Disease is a mostly genetic disease. There have been genetic markers found in the blood that are strongly linked to the condition (HLA-DQ2 and HLA-DQ8). People with Celiac Disease also produce antibodies which show that the disease is 'active' and damage to the villi. The villi are like tentacles in your intestine that wiggle around and absorb the nutrients from food. When these villi get damaged, you struggle to absorb fats and other nutrients and can become very sick. To diagnose Celiac Disease you will have to have blood tests done first. If these come back positive, you'll then be sent for a biopsy of the villi (which isn't pleasant). If these show signs of Celiac Disease, you will be diagnosed. The diagnosis has NOTHING to do with symptoms. You can have no symptoms and still have the condition.
Compared to Celiac Disease, very little is known about non-Celiac gluten intolerance. Basically because most doctors don't care. If there are no serological tests to prove you have a disease, it doesn't exist. Unfortunately this is the case here. But there has been a lot of ground breaking research done in the past year that shows non-Celiac gluten intolerance is a very real condition. If you do nothing else and don't bother to read the rest of my answer, please read this website- http://celiacdisease.about.com/od/glutenintolerance/a/Gluten-Intolerance-Research.htm.
The tricky thing is that you've probably already begun a gluten free diet. So if you have not have the tests for Celiac Disease, unless you want to go back to eating gluten, it's too late. For the doctors to observe the antibodies and damage caused by Celiac Disease, you must be eating gluten for at least 6 weeks before your tests. If you're feeling a lot better now, you most likely don't want to feel crap again! And at the end of the day, it doesn't really matter. The outcomes will still be the same, if you test negative for Celiac Disease, then you'll probably go back to eating gluten free because you feel better, if it comes back positive then you have no choice but to go back on a gluten free diet. If you feel better eating gluten free, then you have your answer.
But the main things you should be aware of, do not screw around with your diet. If you definitely have just a gluten intolerance, you could eat gluten every now and again and just put up with it. But if you have Celiac Disease, this a big no-no! You should also ask your doctor to test your iron, b12, foliate and vitamin D levels. These can be decreased in Celiac Disease. Also, if you're prone to broken bones, then I'd recommend a bone density scan. Osteoporosis occurs in people with Celiac Disease at a higher rate than the general public. Also, people with Celiac Disease may have more problems with their teeth. So regular check ups are very important. And also be aware that having one autoimmune disease, makes you more likely to develop more.
But none of these ^^ problems occur in non-Celiac gluten intolerance, they ONLY occur in Celiac Disease. So this is not advice that you should go out and take/do. You probably don't need a bone scan and it's likely your teeth are fine, but it's just something to keep in the back of your mind.
I hope that's helpful.
....