Assuming your source speaks the truth, different ingredients, fermentation methods, and distillation procedures result in wildly varying beverages. Vodka's fermentation and distillation are generally different from rum, and the resulting spirit tends to be lower in congeners and other non-ethanol substances (unless added after distillation).
Your source, however, is not necessarily correct. Both the FDA and the TTB disagree, and have ruled that you cannot claim as "gluten free" any alcohol beverage made from gluten-containing grains, no matter distillation method you use (bolding mine):
Consistent with the new FDA regulations, TTB will continue to consider “gluten-free” label claims for alcohol beverages that are made from gluten-containing grains to be misleading to consumers who are seeking to avoid the consumption of gluten for health reasons. However, products made from gluten-containing grains may be labeled with a statement that the product was “Processed,” “Treated,” or “Crafted” to remove gluten, if that claim is made together with a qualifying statement that warns the consumer that the gluten content of the product cannot be determined and that the product may contain gluten.
You can read a full explanation on their rationale here.