However, in other choices of foundations, such as Grothendieck universes, there exist both “small sets” (sets that belong to the universe) and “large sets” (sets that do not, such as the universe itself). In this case the adjective really is necessary. We also gain an intermediate notion of moderate set: a subset of the universe, which may be small or large. (Every small set is moderate, but not conversely; again, the universe itself is the standard counterexample.)
Since in many cases the choice of foundations is irrelevant, it makes sense to always say “small set” for emphasis even if one has in mind a foundation where all sets are small.