Wheat is one of the more popular grains containing gluten. "Wheat-free" is not the same thing as "gluten-free". Some people, typically those with Celiac Disease are allergic or intolerant to gluten, which is a mixture of protein fragments found in common grains. Others, diagnosed with a food allergy are allergic to the wheat protein itself. This includes wheat globulins and wheat albumin. If you are allergic to wheat, you will want to avoid wheat, and it's by-products.
Wheat allergies usually present themselves in children, and can be outgrown. A wheat allergy is commonly confused or mistaken with celiac disease. Celiac disease is a digestive disorder, sometimes found to be linked through genetics, that creates an adverse reaction to gluten. Those with this disorder must avoid gluten, that is found in rye, barley, wheat and some oats. People who are allergic to wheat have an IgE-mediated response (immediate, cytotoxic, immune complex, or delayed hypersensitivity reaction). If you have a wheat allergy, you may tolerate other grains.
I hope this helps in understanding the difference between these two commonly mistaken issues and helps keep you safe from mistakes in the future.