First of all, let me start out by saying that I'm not an expert on this subject. What I have written below represents the results of about 5 hours of research in a library, and little more.
Fine, or hard-paste, porcelain consists of Kaolin, a type of clay, feldspar, and possibly other materials. It is fired at about 1400 degress C.
Bone China, on the other hand, is a mixture of this porcelain mixture and about 40 or 50 percent bone ash. The bone ash serves to whiten and slightly strengthen the porcelain, while reducing the necessary firing temperature by about 150-200 degrees C.
Bone china was invented in England in the late eighteenth century, and was apparently superior in strength, translucency, thinness, and whiteness to the porcelain then being produced in Europe.
It is possible to make a lesser grade of porcelain, known as soft-paste porcelain, by incorporating glass into the clay. I don't think any consumer tableware available today is made of this.
In a 1970 book, "Every Woman's Guide to China, Glass and Silver" (Arlene Hirst, Arco publishing, p. 15), I found the following words of wisdom:
Ox-bone primarily serves to whiten the dinnerware. If you compare bone china with all other types of porcelain you will notice the difference immediately. The body of most china has an off-white greyish cast, except for some American china which is ivory colored. True bone china is almost snow-white. Ox-bone does make the body of china slightly harder; however, the glaze is fired at a lower temperature than porcelain so the two factors tend to cancel each other out.
Therefore, I would say that you should probably ignore any minor differences between porcelain and bone china in your shopping, and concentrate on what you like.
I heard from a porcelain collector that most porcelain made today for dinnerware is bone china, though actual bones are apparently no longer generally used; instead they use the pure chemical, calcium carbonate, that is the important ingredient from the bones. (I have no particular knowledge of the correctness of these statements, though the collector did seem to know what she was talking about.)