Answer: Florist (or cutting) mums and hardy (or garden) mums come from the same original parent -- a golden-yellow daisylike mum from China. Today's hybrids in both categories are the results of endless crosses between several species from China and Japan. The result of such hybridization performed over hundreds of years is different types of mums that perform for two distinct purposes.
Florist mums are large-flower plants with many possible bloom forms, from quilled to pompon to spider and more. Grown in greenhouses and used only as indoor plants, florist mums produce few, if any, underground stolens, which are necessary if the mum is to survive cold weather. Florist mums planted outside are most likely being used as short-term bedding plants that will be removed when the blooms are spent. You can plant a potted florist mum you receive as a gift, and it may grow for the summer, but it will not survive the winter, no matter how much protection you give it. Garden mums, on the other hand, produce underground stolens and can survive cold better. Most garden mums are perennials in Zones 5 to 9 and much tougher than florist types. Some cultivars are less hardy than others and can be killed by an early spring frost.
With this said, I have found that if you plant florist's mums in the spring (rather than in the fall) and they have all summer to establish their roots, it is likely that they will overwinter and produce stolons, which will allow them to become perennial in your garden. I think it's worth a try!
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