No, these two terms have a somewhat different meaning when used on food labels.
The term "gluten-free" has been approved for use on food labels since 2013, and it is the U.S. Food and Drug Administration (FDA) that sets standards for its use. When you see "gluten-free" on a food label in the U.S., that food must conform to FDA standards for labeling. The basic components of FDA labeling standards for "gluten-free" are as follows:
In practical terms, 20ppm of gluten proteins in a food would mean 2 milligrams of gluten proteins in 100 grams of the food (about 3.5 ounces). For most people with healthy immune, inflammatory, and digestive systems who nevertheless experience gluten-related problems, this trace-level amount of gluten appears to be well-tolerated. However, for individuals with compromise to these body systems, for example, persons diagnosed with celiac disease, there is no guarantee that unwanted reactions will be prevented following intake at this trace level. It is also worth noting here that consumption of multiple gluten-free products throughout the day could result in exposure to substantially higher milligram amounts of gluten proteins in foods that fully meet the FDA guidelines for "gluten-free."
As an FDA-approved labeling term, "wheat-free" means that wheat (as a species of plant) as well as spelt, kamut, and triticale were not used in production of a food. "Wheat-free" also means that derivatives of these plants were excluded from a food's production. The FDA identifies exact species of wheat that cannot be used in food product if a company wants to label those foods as "wheat-free." The plant species listed by the FDA are: common wheat (Triticum aestivum L.), durum wheat (Triticum durum Desf.), club wheat (Triticum compactum Host.), spelt (Triticum spelta L.), semolina (Triticum durum Desf.), Einkorn wheat (Triticum monococcum L. subsp. Monococcum), emmer wheat (also known as farro) (Triticum turgidum L. subsp. dicoccon (Schrank) Thell.), kamut (Triticum polonicum L.), and triticale (x Triticosecale ssp. Wittm.). In this case of wheat-free labeling, what kamut and spelt and triticale all have in common with wheat is the Triticum genus of plant to which they belong, rather than the presence or absence of gluten proteins in their chemical make-up.
The label "gluten-free" restricts the amount of gluten proteins in a gluten-free food to 20ppm or 2 milligrams per 100 grams of food. For persons needing to avoid all gluten proteins in their meal plan, this labeling term is not sufficient to guarantee that outcome. For most people with healthy immune, inflammatory, and digestive systems who nevertheless experience gluten-related problems, however, this trace-level amount of gluten seems to be well-tolerated, provided that a person does not consume too many different gluten-free products throughout the day, or overconsume any particular gluten-free product. The "wheat free" label does not guarantee an absence of gluten proteins in a food, since "wheat-free" only restricts the presence of grains belonging to the Triticum genus of plants (wheat, spelt, kamut, and triticale). Rye and barley, for example, are not included in the "wheat-free" definition since they do not belong to the Triticum genus. (Rye belongs to the Secale genus and barley belongs to the Hordeum genus.)
If your reason for eating "wheat-free" is to avoid gluten proteins, you are better off choosing "gluten-free" foods than "wheat-free" foods (even though the "gluten-free" label is not foolproof since it allows for trace amounts of gluten proteins). "Wheat-free" is a good labeling choice if you are trying to avoid wheat as a plant component in your food.
To see the research articles we reviewed in the writing of these articles, see here.