What is the difference between the chemicals that a professional pool supply store sells compared to a big box retailer?

A big box retailer will not stock the variety of chemicals that a professional pool supply store can provide. You may be able to choose from several different brands at a professional pool supply store. Also, most of the highest quality brands can only be found through authorized dealers and NOT in a mass merchant.
Unlike the big box retailer, the pool store will be able to test your water (this is usually a free service) and recommend solutions for possible problems with clarity. If you build a relationship with the staff at a pool store, they will help you to have a clear and safe pool for your family.