Modern wheat is a genetically modified grain that is nothing like heritage/ancient grains. It has a gluten that people are not used to digesting. To be diagnosed for Celiac, generally you need to have antibodies to the gluten (which if I understand correctly means that your intestines are so bad off that the proteins entered your blood), but a lot of people can be sensitive to gluten without it showing up as antibodies in the blood.
I have been going mostly modern wheat free since November, and it's made a big difference in both my mood and how much sleep i need. I've gained about 2 functional hours a night (used to sleep 9-10 hours, now I only need 7-8 hours), and I don't get super cranky right before bed (which used to involve screaming at kids a lot). I have substituted other carbs for wheat, so it's not that I'm suddenly going paleo/low carb. I also find that I'm on the computer less because I feel less lethargic. My son was also doing a low-wheat diet and dropped 10lbs in a couple of months. He stabilized at a really good weight/BMI.
Wheat is in sooooo much stuff, that eliminating it entirely is pretty tough, and frankly a lot of gluten-free stuff doesn't taste as good. I don't like to buy a lot of packaged foods, but we do buy Glutino cookies and Trader Joe's gluten free brownie mix so that I don't have to give up every single sweet except for ice cream. A couple of times I've ended up looking for stuff at Whole Foods, though I hate shopping there.
Some people say that to really heal from gluten damage, you need to be off wheat entirely for 3 months. I don't know--maybe I could feel absolutely fantastic if I did, but for me, I limit it to once or twice a week and call it a good compromise.