In case you aren’t interested in the details but you are interested in a quick question that clears up whether you are getting a legitimate crystal simply ask if it is a variety of mineral and if so, what variety? If it isn’t a variety of mineral, it’s a rock. If you want to verify, the best place to do so is: https://www.mindat.org/, the world’s largest public database of mineral information.
The crystal market is loaded with stones and crystals that aren’t verified on any “established” list, examples of these are boji stones, lemurian crystals, Columbian amber and azeztulite. Common minerals and rocks are often given trade names, like a brand name. If you see a TM next to the name of a crystal that is what it is, not a unique mineral variety.
As to the differences:
CRYSTALS: Natural or manmade they are either transparent or have “external planar faces” or flat sides, like pyrite. Most don’t have names, they are called by their mineral variety name such as ‘amethyst crystal’ or ‘quartz crystal.’ There is no ‘established’ list of crystal names.
MINERALS: The word mineral refers to a non-organic substance naturally occurring in nature as a solid. A mineral has a definitive chemical composition and an orderly crystalline structure of molecules, examples are iron, quartz and carbon. For the sake of clarity here are some things that are not minerals; Pearl comes from an organic source, an oyster and amber originated in a tree so they are not minerals. Oxygen is not found in nature as a solid-not a mineral. Glass has no crystalline structure and granite is a mixture of minerals such as quartz, feldspar and mica, which makes it a rock. There is an ‘established’ list of minerals and their common varieties, they all have formal, universally accepted names via the International Mineralogical Society.
ROCKS: A rock is an aggregate of one or more minerals and maybe some organic compounds. There is no “approved” list of rock names and new ones seem to pop up all the time.
STONES: The word stone is a slang word that refers to a rock or mineral that is not a crystal, like alabaster.