Are they called stones or gems? What's the difference?

Stones. Gems. Rocks. Crystals. A gemologist may differentiate between them. A rockhound may differentiate between them.  An energy healer may differentiate between them. In my world these are interchangeable terms all referring to a hard substance that is of the earth. I enjoy the phrase "Fine Gemstones" because I choose what I consider the finest of the exquisite cabochons available to me when I am at a wholesale gem show making my choices. Attending a gem show for inventory purchases I call "Rock Shopping". 

On a related topic, I have used the vocabulary word: cabochon.  cab·o·chon [ˈkabəˌSHän] noun a gem polished but not faceted.

While I mostly use gems cut in cabochon form, I do use some faceted gems as well. A faceted gem example is a diamond. There are many terms for defining specific cuts that designate their shapes and style of faceting, but I am not well-versed in these.