Here's the hard part. It's vastly different from person to person.
My chiropractor has a gluten allergy of sorts, but it doesn't seem too severe to him. Whenever he has any, he gets a migrane throughout the next day and it will then pass with no other ill effects.
One of my coworkers has had to go gluten free to alleviate arthritic pain since it causes immflamation of his joints.
As for me, well. From what I've determined so far, it causes issues with my digestion so that I don't seem to absorb nutrients leaving me constantly hungry and the food itself seems to pass through without being digested. (I call it the carrot test. If I eat a carrot and don't digest it, I'm having problems.) It's not quite the runs, but 75% there. This lasts for a week. Then for the next two weeks I go into, I guess a sort of chemical withdrawl where I'm hazy, easily irritated, and have a lot of trouble concentrating.
I've also had light ezecma (dry skin rashes) frequently prior to being gluten free which may be connected since they didn't go away permanently until I cut it out of my diet. And I may have had tooth enamel problems during the time, but that's harder to tell. I used to get cavities almost every checkup despite frequently brushing and flossing. These past few years I've been forgetting to brush a bit, but haven't had significant problems like before.
Some people have also said that there was some bloating and that my face doesn't seem as swollen, despite not having much of a weight loss based on excluding gluten. (Many gluten free foods are more processed and have a higher Glycemic index too).
I'll see if a thyroid problem was related the next time I have a blood test, but that may be a separate item.
I've only been gluten free for about 3-4 years now, (Which does mean I miss Guiness and other stouts and red ales terribly.) I was self diagnosed in that I had stomach issues, including lactose intollerance (which wasn't too bad since yes, I had the runs for that, or the next day, but nothing much worse than the gluten thing). I was ramping up some whole wheat baking. (Bread every other day, cakes etc) and kept getting worse so when I moved and got a different job, I switched my diet and went "Crap. It's working." Having to deal with a month of issues and not just a single day means I avoid it more than say... my chiropractor does.
Some of the others in the Gluten free forum have different symptoms and have to either be gluten free, or make gluten free beer for others (usually the SWMBO).
http://www.csaceliacs.org/celiac_symptoms.php is a good symptom site to see some of the other issues some people have.