Stains are consistant, stable, safer and are predictable. They work hard to give results. However, they do not dissolve well in glaze and are generally more expensive than oxides.
Rare Earth Elements are lanthanide oxides. These metals possess special effects such as fluorescent and dichroic properties, and are active under multiple waves of light.
Oxides have a wide range of possibilities and variables. Thus making them difficult to predict. However, due to their low melting point, they are useful for tinting clear glazes.