In the past, precious stones refered to Diamond, Ruby, Sapphire & Emerald. Semi-precious refered to all other gemstones. Although these terms are still used by consumers and some jewelers, Gemologists do not differentiate gemstones along the "precious" "semi-precious" line because it is grossly inaccurate.
For instance, tourmalines are usually considered "semi-precious". However, a fine Paraiba tourmaline is more expensive than many rubies and sapphires. In this day and age, the "precious" "semi-precious" classification is meaningless.
To answer your question: There is no difference. ALL gemstones are precious. Amethyst is generally more expensive than blue Topaz because of a higher market demand for it. Both are not particularly rare, although Amethyst is possibly the rarer material.
Prices differ because of a variety of factors. For instance, rarity, market demand, marketing, treatments available and accepted, fashion etc.
Rarity on its own does not guarantee that a gemstone will be more expensive. Market demand is equally important. Have you heard of Painite? Taafeite? Will you pay "big money" for them? The answer is probably "No". The reason is not only because they are rare and are not available in large/commercial quantities, but also because the market doesn't know them and therefore there is low demand for them.
Hope that helps!
I LOVE gemstones and I teach gemology.