Having used both at the same time, I have noticed that cardboard boxes on the floor of the closet are more likely to pick up odors (mustiness) than plastic tubs. I have even stored fabric in a plastic tub OUTSIDE, while the sewing room got painted, and forgotten it for over a month without the fabric having any problems.
I do prefer the flexibility of cardboard boxes for cramming in more fabric in a limited space but plastic keeps it cleaner and (I think) neater.