So Rivulets actually really neat because external gills tell us a ton of things: Rivulets most likely from a tropical/warm water location with murky water, such as a swamp or bog. We can guess at the murky or low-visibility water due to the eyes. In environments like that animals generally either reduce their eyes and attempt other forms of sensory input, or improve it to make up for the harsher environment. Slugcats, even Rivulet, are not restricted to the water and are highly visual animals- thus the increased eye size can be seen as a corrective measure for their native environment in place of giving up their existing visual strength and terrestrial advantage.
We can infer they likely aren't cold-water because thin, branching structures like external gill fringes would be extremely prone to freezing and lose a ton of heat from their high exposed surface area, which they need for a decent amount of gas exchange at all. Rivulet's fringes are also quiet long, which suggests they are of functional use and need the additional surface area, as many species who are in colder or more highly oxygenated waters have reduced gill size in order to minimize risk. Cold water has higher oxygen capacity than warm water- which means less gill area is needed in order to meet demand. Interestingly, axolotls, which are likely Rivulet's inspiration- are actually cold water animals with notable gills. However they are also much lower energy animals and highly temperature sensitive to maintain metabolism. Given Rivulet's status as a traveler, they likely do not suffer from such a strong need for temperature stability, especially being semi-aquatic and highly active.
As for external vs. internal, internal gills are on average much better for the 'job' of gills, especially in open water, with the additional advantage of being much more damage-resistant (because of this, external gills are mostly used in larval stages where the gill will be reabsorbed and regenerated) but interestingly… its actually way more efficient to breathe air than try and filter it out of the water! This is why whales and other land-animals-turned-aquatic keep their lung systems. We actually know Rivulet themself actually doesn't even truly breathe underwater like a fish does, they can still drowning in-game, but can maintain themselves longer than normal slugcats. This suggests that their gills aren't super core to their overall respiration, since its unlikely that all sources of water in Rain World, including Shoreline, are just so incredibly oxygen poor that Rivulet's gill system is struggles to function as intended in them. Its fairly likely that Rivulet might have some sort of lungfish-like system with access to both functional gills and terrestrial lungs, and is more reliant on their lungs.
So where do long, fragile external gills pay off? In 'bad' water where there is very little oxygen or very little movement (of water over the gills). External gills can be waved from a static position, which is ideal for minimal energy waste or ambush predators. This makes them great adaptations for things like muddy ponds or small flooded areas, especially for animals like salamanders or frogs who will eventually 'outgrow' these areas. This seems pretty fitting for the rain filled waters of Rain World, many of which probably contain large amounts of run off or are stagnate, isolated ponds without much water flow.
This obviously wasn't part of the original, but here is a diagram from a scientific paper about external gills, showing gill variation based on environment. In general the better the water quality (oxygen levels/movement), the less gill you need. Big gills on Rivulet suggests that their normal water is rather hard to breathe in, which would probably explain the lack of Rain World fish.
Also for another fun bonus, many amphibians can do gas exchange with their skin and use this as a means of breathing. Rivulet could be breathing through nose, gills and skin! Yummy Air!!!
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