Wheat Allergy But Not Celiac

This works just like any other allergy.  When the allergen wheat is
ingested, white blood cells send out antibodies to attack it, which
then causes symptoms.

Wheat is a primary source of gluten, which is why this allergy is
often lumped with a gluten free diet.  However,  people with
wheat allergies can eat gluten from non wheat sources. 
This allergy is one of the top eight food allergies and can be
outgrown as a child gets older.