Fabric allows the quilt to breathe, which means that moisture is not trapped inside. (Trapped moisture can cause mold and other fiber deterioration.)
However, it's really the soft plastics that are not so good for quilt storage as they can outgas chemicals that damage fibers. Another problem with garbage bags specifically is that quilts have inadvertently been mistaken for garbage and gotten tossed.
Hard plastic containers are safe except for the moisture trapping issue. To deal with that, some people drill a couple of holes in the container but it's easier to just include some silica gel packets similar to those found in pill bottles, shoes, etc. You can even buy silica gel in bulk at places such as Walmart; it is used to dry flowers.
Edit: I just want to add that if there is any possibility of mice getting into the house, the hard plastic container with silica gel would be the safest. Mice will chew quilts and use fibers for bedding. In this case, you also wouldn't want to drill holes in the plastic bins as mice can squeeze through incredibly small places.