How do we identify the difference between precious and semiprecious gemstones?

Precious means pricey. That is really the only distinction.

There are a lot of surprises in the subject. A ruby is worth a lot more than a diamond of equal quality. Good emeralds are almost impossible to find any more. There are so many diamonds that the cartel has to hold most of them back to keep prices up, and there is no resale market at all. Diamonds are all less than one carat because they crush the ore to that size at the mine. Most imitations of diamonds look better than genuine stones. Amethyst (purple) and citrine (yellow) are chemically the same stone. One can be changed to the other by baking, but it can't be changed back.