Why is Rum not affected by distillation like Vodka is?

There's 2 different statements there:

1. Distillation makes alcohol gluten free
2. Rum is gluten free but high in FODMAPs

Both could be independently true or false, there's no logical tie between the two. Some of the information in that article is incomplete and/or wrong:

• Distillation does not result in gluten free alcohol in all cases. It reduces gluten a substantial amount but there is frequently traces left which may not be tolerated. The standard I've heard for gluten free is less than 20 parts per million, but that 19ppm may still cause a problem
• Gin and Vodka are made from wheat, barley or rye. This isn't always the case, any sugar-producing plant can be used for gin and vodka, for instance beets, potatoes, grapes, sugar cane, corn, etc. There's plenty of grape and potato vodka and gin, which are gluten free because the sugar source never had any in the first place
• Not all whiskey is made from wheat, barley or rye. Some whiskeys like bourbon are made from corn, although the requirement is only 80% so some may contain wheat as well

Repeated distillation can produce a pure alcohol, however it would remove the compounds that give spirits their flavor. Alcohol stills used to make drinking spirits are often shaped in specific ways to retain flavor rather than get rid of it. Some of these flavor compounds aren't friendly to people with stomach problems, so the less flavor the better essentially if you have issues. It helps of the sugar source is bland to begin with and there's no flavor additives. Rum has lots of flavor after distillation because sugar cane has flavor, whisky has flavor from the grain, but has more added if it's smoked (Scottish whiskeys are peated) and gets further from the barrels its aged in. Potatoes are comparatively bland, and vodka isn't generally barrel aged (it is available but it's not the clear stuff you know) so it's a clearer and "cleaner".

Some rums contain sugar because the producer adds it after distilling, presumably because it improves the taste, which (AFAIK) is rather unheard of for other spirits. Some non-authoritative sources:

https://vinepair.com/wine-blog/does-rum-have-sugar-in-it/

http://www.alcademics.com/2014/01/how-much-sugar-is-added-to-your-rum.html

http://refinedvices.com/forums/viewtopic.php?f=4&t=485&p=2515#p2515

Note that not all rums contain added sugar.

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.