The difference between earthenware and stoneware is firing temperature, and everything else follows from it: how porous the fired body is, what colours are available, how strong the ware is and what it costs to fire. Choosing between them is the first decision a potter makes.
Temperature and porosity
Earthenware fires around 1080 to 1180 degrees and remains porous, so the glaze is what keeps water out. Stoneware fires around 1180 to 1300 and vitrifies, so the body itself is largely non-porous and a chip in the glaze does not make the pot leak. That single difference is why most functional studio pottery is stoneware.
Colour and palette
Earthenware's lower temperature preserves colourants that shift or burn out higher up, which is why bright reds, oranges and clear yellows are far easier at earthenware temperature. Stoneware's palette is broader in the muted and reactive directions and narrower in the bright ones. Traditions grew around both, which is a better guide than any generalisation about quality.
Strength and use
Stoneware is harder and more chip resistant, and vitrified stoneware survives dishwashers and daily use readily. Earthenware is softer and more vulnerable at rims and edges, and depends on an intact glaze. For tableware in constant use, stoneware is the practical choice; for decorative and traditional work, earthenware has advantages that are not about durability.
Cost
Earthenware bodies are generally cheaper per kilogram, and earthenware firings cost less because the kiln climbs less far and holds for less time. Over a lot of firings that adds up, which is one of the reasons schools and classes often work in earthenware. The clay difference is smaller than the firing difference.
Questions people ask about earthenware
Which is stronger, earthenware or stoneware?
Stoneware. It vitrifies at higher temperature so the body itself is hard and largely non-porous.
Is earthenware waterproof?
The body is not; the glaze is what keeps water out. That is why a craze or a chip matters more on earthenware.
Which is cheaper?
Earthenware, both in clay and in firing, because the kiln does less work. The firing difference is the larger one.