The reason for the different terms is the same as the reason for the different terms "linearly independent set" and "basis".
Every linearly independent set is a basis for the subspace it spans. But when working in a larger space "basis" means "maximal linearly independent set" (not just spanning a subspace but spanning the whole thing).
An orthogonal set (without the zero vector) is automatically linearly independent.
So we have "orthogonal sets" and then maximal ones are "orthogonal bases".
Note: In the end we're essentially just tacking on the adjective "orthogonal". We don't keep the words "linearly independent" in "orthogonal linearly independent set" because they're redundant.