In common parlance the stories of the Greek and Roman deities and heroes are indiscriminately referred to as myths and legends. If we wish to be more careful, however, we can differentiate between the two types of story, and between them and folktales and fairy tales, although a story may shift between these different categories, or may contain elements from each of them.
Briefly, we can say that a myth gives a religious explanation for something: how the world or a particular custom began. There is usually no attempt to fix the myth into a coherent chronology related to the present day, though myths or a cycle of myths may have their own internal chronology. The story is timeless in that the events are symbolic rather than just the way it happened.
In calling a story a myth we are expressing no opinion about whether it is true or not. In the days, when, at least publicly, Christianity was assumed to be true and other religions false by those writing about religion (say, the 19th and early 20th centuries), the specialists' use of the word myth was closer to the popular use to mean an untrue religious story, and it was only used for other people's religion. As anthropologists and students of religion came to take a more impartial view of the world, it was recognised that certain Christian stories shared many of the features of myth, and could be called myths if the idea that a myth was necessarily false was shed.
A legend, on the other hand, is a story which is told as if it were a historical event, rather than as an explanation for something or a symbolic narrative. The legend may or may not be an elaborated version of a historical event. Thus, examples of legends are the stories about Robin Hood, which are set in a definite period, the reign of Richard I of England (1189-99), or about King Arthur, which were perhaps originally based on the exploits of a Romano-Celtic prince who attempted to resist the expansion of the Anglo-Saxons in what was to become England. The stories about Robin Hood and King Arthur have been elaborated and expanded on down the years.
While myths and legends may be transmitted orally or in writing, folk tales tend to be transmitted orally, and although they are transmitted from generation to generation and so their origin or author is unknown, they are more definitely felt to be stories, i.e., fiction. Many European folktales were written down in the 19th century, and some at least were transformed into fairy tales, which tend to be more consciously literary productions with a definite author, such as Hans Christian Andersen. Typically, folk and fairy tales involve magic and magical creatures and people such as witches, dragons and dwarves rather than religion. Examples are Cinderella and Jack and the Beanstalk.
Let's look at some examples, now, from the ancient world. First the myth of the Abduction of Persephone.