The term "fiber base" refers to papers produced from either alphacellulose sources (e.g., wood pulp) or cotton fiber source materials. When properly manufactured to be acid neutralized and ligin-free, the final cellulosic content in the paper base sheet is chemically quite similar if not identical, so it's unclear that cotton sourced papers will necessarily outlast papers derived from other cellulosic fiber sources. History has already proven both types of base sheet can have excellent durability over time. The key distinction in photographic, inkjet, and other paper-based media then becomes what sizings or coatings also get used to produce the final product. For example, in the late 1960s, photographic manufacturers began to introduce polyethylene coatings coated on both sides of the base fiber sheet upon which additional silver halide sensitized photographic gelatin coatings were then applied to enable rapid processing and drying. These papers are referred to generically as RC (resin coated) papers and distinguished from "traditional fiber base" photographic papers that had been around for well over a century. Note the photographic industry rarely if ever used cotton as the base sheet, so most photographs that are not on RC papers are in fact produced from highly purified wood pulp not cotton fiber source. Because the modern RC coated base sheet materials are widely available and consumers got used to the more modern "look and feel" of RC papers, inkjet coating formulations were then logically adapted to these RC base media as well, and hence we have all the classical distinctions between RC and Fiber base even in this modern inkjet age with the additional option for cotton fiber as well as wood pulp fiber. Again, either source of cellulosic component can produce a highly refined and chemically stable base sheet. It's the subsequent coatings and inks on top that tend to limit the longevity. That said, cotton papers do tend to have slightly different mechanical properties, usually being softer and more supple in hand than other slightly stiffer alphacellulose derived papers. Hence, many artists prefer the full cotton "rag" papers for feel-in-hand properties plus they have been influenced by the industry promoted association of cotton paper being only of highest quality whereas "wood pulp" papers can vary all over the map in quality (from bad to as good as cotton depending on how the material is refined during manufacture).
Note: my comment above was prompted by a prior question/remark in this thread, but that remark has since been edited out. I could therefore sensibly remove this reply, but I think it may be of some general value to this post, so I will let it stand.
cheers,
Mark