Gluten: Gluten is the protein that is present in wheat, barley and rye. If you are told to avoid gluten, avoid foods like pasta, breakfast cereals, pastries, other flour-containing foods, such as pizzas and certain sauces, and oats.
Wheat: People who are not gluten intolerant or coeliac may still find comfort in avoiding wheat, but they can still tolerate oats, barley, rye and so on. This is when wheat, not gluten is the cause of any health issues. Try, for instance, having the occasional slice of rye bread to see how your body reacts to it.