Pottery and ceramics are pretty much the same.
Both are general terms that describe objects which have been formed with clay, hardened by firing and decorated or glazed. The difference comes mainly from the temperatures at which the clay is fired.
There are three main types of pottery/ceramic: earthenware, stoneware and porcelain.
Earthenware
Earthenware is clay fired at a relatively low temperature - between 1,000 to 1,200ºC (1,830 to 2,190ºF), it can be fired effectively as low as 600ºC. This results in a hard but brittle material which is slightly porous. Basic earthenware, often called terracotta, is not waterproof and absorbs liquids.
Earthenware could be coated with a ceramic glaze and fired in a kiln for a second time make it waterproof.
Stoneware
Stoneware is made from a particular clay which is fired at a higher temperature of 1,100 to 1,300°C (2,110 to 2,370ºF) resulting in a stronger material, with a denser, stone-like quality. The finished product is waterproof and unlike earthenware, does not need to be glazed and fired for a second time.
Porcelain
Porcelain comes from a refined clay which is fired at high temperatures of approximately 1,200 to 1,400°C (2,200 to 2,600ºF) resulting in an extremely hard, shiny material often white and translucent in appearance.
The earliest forms of porcelain originated in China around 1600BC and this association popularized the term 'fine china’, or bone china.
Bone china porcelain is made when ground animal bone is added to the clay, in order to create an even more durable material.
Summary
The main points of comparison between Earthenware, Stoneware and Porcelain, will be the temperature at which the clay is fired and the resulting strength, water resistance and durability of the finished products.
The quality of the finished product will be dependant of the quality and purity of the clay used to create them, but as a general rule, Stoneware and Porcelain will be the two more durable forms of ceramic, which are commonly used as tableware.