In some choices of foundations, one says small set in order to amplify that one really means a set and not a proper class. Strictly speaking, in this case one could just say “set”.
However, in other choices of foundations, such as Grothendieck universes, there exist both “small sets” (sets that live in the universe) and “large sets” (sets that do not). In this case the adjective really is necessary. 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.