Well, "gemstone" refers to a stone that is cut and polished, and can be set into jewelry. "Crystal" has two meanings. The better one is the natural form of gemstones (how they come out of the earth). A secondary meaning of "crystal" is a glass gemstone.
If you are seeing a description of the stone in a piece of jewelry and the seller is calling it "crystal", you can be sure it is glass.
But as Tourmaline said above, the natural form before cutting is called crystal as well.
Crystals are formations that replicate themselves in regular patterns. For example, in the right circumstances, a carbon molecule will bond with other carbon molecules in a hexagonal ring, and then more rings will form with each ring sharing a side with its six neighbors. The rings are at different angles to their neighbors (with the shared sides) so the whole thing turns into a three dimensional net. Which is to say, a diamond.
Not all crystals are what we would call gems. Gold is a crystal for instance. Drywall is a crystal. Crystals do tend to be pretty because their uniform structure causes them to reflect light in a smooth sheet, or a little pile of sheets, which makes them shiny or glittery.
Yes, crystals normally refer to the natural state of a gemstone (before cutting and polishing). Gemstone usually is for a crystal that is cut and polished and of a standard and variety suitable for use in jeweler. Not all crystals are cut and polished for gemstone use.
Crystal also sometimes refers to glass or manmade “sparkly” stones used in costume jewelery. For example Swarovski crystals, they are glass.