I think most of the time (with the exception being people who have Celiac's disease), those who get legitimate improvements from cutting wheat out of their diet are merely feeling the effects of less carbs, or just eating less in general.
As someone who has a passion for cooking (including baking bread), I'm obsessed with nutrients and their effects on on physiology.
As someone with obsession problems and annoying attention to detail, I'm also fascinated with epidemiology and the "post hoc ergo propter hoc" logical fallacy.
I've found improvements from tracking my nutrients with a smartphone app called MyFitnessPal, and working out. Most of the population in the US eats too many carbs, too much fat, AND more protein than necessary. Just eat too much in general.
I also think when we make a change in our diets, it gives us structure, and structure is something that those on the spectrum require, right? I think creating rules to follow (be it diet, exercise, or practicing something) forces us to pay attention to ourselves and create vast improvements in our lives.