A mushroom is scientifically defined as one of a family of land-borne fungi that grow in moist conditions and are sometimes edible. Mushrooms are normally associated with the standard cultivated mushroom shape of a stem and a cap, but many wild mushrooms have many different appearances. The word "toadstool" is somewhat older than the word "mushroom," dating back as early as the 14th century. Although the differences between toadstools and mushrooms have been variously defined over the years, "toadstool" is not a scientific term and there is no formal difference between a toadstool and a mushroom. The best that can be done in identifying the differences between toadstools and mushrooms is to look at several different kinds of erroneous definitions that have variously been used over time.
The word "toadstool" and its many variants were originally used to describe a type of mushroom on which a small toad may sit. Under this definition, a toadstool has a stem and a cap wider than the stem. Today, this is in fact the appearance of most cultivated mushrooms sold in stores, although mushrooms found in the wild may have several other shapes.
The exactly opposite definition is sometimes used, saying that a mushroom has a stem and a cap while a toadstool may lack a stem, lack a cap, or have a blended shape in which the stem and cap are difficult to tell apart. This is also a colloquial layperson-use definition rather than a scientific one.
Another way that toadstools have been distinguished from mushrooms is by saying that toadstools are poisonous but mushrooms are safe to eat. There is, however, no correlation between the shape or coloration of a mushroom and whether it is safe to eat, poisonous in various degrees, or hallucinogenic. Many mushrooms with the classic toadstool shape are not safe to eat, while many others are; and mushrooms of a stemless shape, for example, may be safe or unsafe to eat as well.
Toadstools are often associated with fairy tales and stories in which people are poisoned from eating them. In European folklore they have also been associated with gnomes. The toadstool associated with gnomes has a stem and a cap wider than the stem and often a red polka-dot coloration. Many mushrooms share that shape and have similar or different coloration, but the word "mushroom" is generally not associated with fairy tales, gnomes, or folklore.
In short, there is no scientific difference between mushrooms and toadstools. When referring to the type of fungus normally referred to as a mushroom, the words "mushroom" and "toadstool" may be used interchangeably.