Mythology has many elements to it, but here I'll focus on two. First, it develops over a long period of time. It's sometimes compared to the game of telephone, where one person whispers something in someone else's ear, the second person whispers to a third, etc. After several people, the story has become mangled. This, however, is incomplete. A closer parallel would be the same game where every third or fourth person has to say what he heard aloud, and allow himself to be corrected by the first person. So with mythology: it takes a long time for it to replace the original story because the original is still available and has more credibility.
The telephone game analogy suggests that mythology evolves slowly over time. It should be noted, however, that the inaccurate ideas may arise quickly. What takes a long time is the replacement of the original with the myth. The collective memory of the actual events simply takes a long time to dissipate. A. N. Sherwin-White argued in Roman Society and Roman Law in the New Testament that two or three generations was too short a time to have the original story replaced by a myth. Indeed, when it was first suggested in the 19th century that the accounts of Jesus in the Bible are mythological, it was assumed that none of them were written until the late 2nd century, since that's how long it would have taken for a myth of that magnitude to arise and be widely accepted. At least there aren't any known examples of it happening faster. Indeed, were this not the case, we would virtually have to abandon the field of ancient history, since almost no ancient historical writings were written close in time to the events they narrate. Since all but a few of the books of the New Testament are dated by scholars to within the first century, the time necessary for them to be mythological simply isn't there. In fact, there is no competing story other than the one found in the gospels until you get to the mid to late second century. As William Lane Craig writes:
The letters of Barnabus and Clement refer to Jesus’ miracles and resurrection. Polycarp mentions the resurrection of Christ, and Irenaeus relates that he had heard Polycarp tell of Jesus’ miracles. Ignatius speaks of the resurrection. Quadratus reports that persons were still living who had been healed by Jesus. Justin Martyr mentions the miracles of Christ. No relic of a nonmiraculous story exists. That the original story should be lost and replaced by another goes beyond any known example of corruption of even oral tradition, not to speak of the experience of written transmissions. These facts show that the story in the Gospels was in substance the same story that Christians had at the beginning. (emphasis mine)
A second element of mythology is that it functions as a literary genre. This is a very important point: as the story changes, so does the way it is told. To suggest that the ancients could have written mythology but not in the genre of mythological writings is simply incoherent; these were two aspects of one thing. It is only in the Modern era that we have classified these literary genres and how they function. So in order for someone in the ancient world to write a mythological story but not in the mythological genre is to suggest that he foresaw the development of Modern literary criticism and adjusted his style of writing in order to trick his future readers -- two millennia in the future -- into thinking that the stories he was telling were not mythological when they really were. This is about as conspiracy theory-ish as you can get without spontaneously combusting.
One aspect of the process of mythologization is that it tends to eliminate irrelevant details -- either by simply erasing them or by ascribing some meaning to them (thus eliminating their irrelevancy). In a myth, every element has a role to play, but historical writings record things that are "messy", that don't have some meaning to the overall story. The biblical accounts of Jesus are replete with such little details. Several times before Jesus would speak to people, Mark records him sighing deeply (7:34; 8:11-13) or gazing at them intently (3:5, 34; 10:23). When a crowd brings an adultress before Jesus, he stoops down and doodles in the dust with his finger (John 8:2-11). A few copies of the New Testament several centuries later tried to accommodate this by adding that Jesus wrote down the sins of the woman's accusers to show that they were not without sin. That's exactly how mythology works, by changing the details so that they have some relevance to the story.
Gregory Boyd gave several examples of this in John 20:1-8 in a letter he wrote to his non-Christian father, later published as Letters from a Skeptic (I should note that I disagree with Boyd on some of the points he makes here):
Early on the first day of the week (when? does it matter?), while it was still dark (who cares?), Mary Magdalene (an incriminating detail, see the next criteria) went to the tomb and saw that the stone had been removed from the entrance. So she came running to Simon Peter and the other disciple, the one Jesus loved (John's modest way of referring to himself -- another mark of genuineness) and said, "They have taken the Lord out of the tomb, and we don't know where they have put him!" (note her lack of faith here) So Peter and the other disciple started for the tomb. They were running, but the other disciple outran Peter and reached the tomb first (John's modesty again, but who cares about this irrelevant detail?). He bent over (the tomb entrance was low -- a detail which is historically accurate for tombs of wealthy people of the time -- the kind we know Jesus was buried in) and looked in at the strips of linen lying there but did not go in (why not? irrelevant detail). Then Simon Peter, who was behind him (modest repetition again), arrived and went into the tomb (Peter's boldness stands out in all the Gospel accounts). He saw the strips of linen lying there, as well as the burial cloth that had been around Jesus' head (irrelevant detail -- what was Jesus wearing?). The cloth was folded up by itself, separate from the linen (could anything be more irrelevant, and more unusual, than this, Dad? Jesus folded one part of His wrapping before He left!). Finally the other disciple, who reached the tomb first, also went inside (who cares about what exact order they went in?).
The presence of little details like this should not be understood as absolute. Fully mythological stories can have irrelevant details, and historical writings can show how little details were actually relevant to what was going on. The point is that in general, the more such details there are, the less mythologized the story is. This gives us the ability to test how far along the mythologization process a story is.
Here's a non-biblical example: The Voyage of Saint Brendan is an early medieval text describing an Irish monk who built a small leather boat and, essentially, sailed it around the North Atlantic Ocean. Tim Severin, in The Brendan Voyage, relates how his wife, an expert in medieval literature, thought that The Voyage of Saint Brendan was a partially mythologized story of something that actually happened.
"There's something odd about the Saint Brendan text," remarked my wife Dorothy one evening. Her casual comment immediately caught my attention.
"What do you mean by 'odd'?" I asked her.
"The text doesn't match up with much of the other literature written at about the same time. The best way to explain it is that it doesn't have the same feel. It's a curiosity. ... The story has a remarkable amount of practical detail, far more than most early medieval texts. It tells you about the geography of the places Brendan visits. It carefully describes the progress of the voyage, the times and distances, and so forth. It seems to me that the text is not so much a legend as a tale that is embroidering a first-hand experience."
Severin decided to build a leather boat out of the same material that would have been available in that particular part of Ireland at that particular time and sail it across the North Atlantic (à la Kon-Tiki). Not only did he successfully sail from Ireland to North America (via the Faroes and Iceland), he learned that a leather boat had great advantages over wooden ones: at one point, they struck an iceberg strong enough that it would have punched a hole in a wooden boat, big enough to sink it. A leather boat, however, can be sewn up en route.
Anyway, the point is that no scholar has ever suggested that the gospels are written in the genre of mythology. Those who have argued that they are mythological (primarily in the late 19th century) said they should be understood this way despite the genre in which they are written. In fact, this is so blatant, so screamingly obvious, that you can verify it yourself: simply read the gospels side-by-side with actual mythological writings -- not modern retellings of mythological stories, but the actual myths themselves. It's obvious that they're not in the same genre. Until fairly recently, it's been a contentious point what genre the gospels belong to, other than that they were roughly historical writings. But in the last few decades, scholars have accepted that they are written in the genre of ancient biography, similar to Diogenes Laërtius's Lives of the Eminent Philosophers. As I pointed out here, that doesn't mean that they are historically accurate in every detail, but it certainly makes it very difficult to claim that they are inaccurate in their central claims.
I've spent an inordinate amount of time on mythology. Urban legend is simpler: it basically lacks many of these elements. An urban legend is not based on a long process of mythologization but on someone telling a false story. Thus, in contrast with actual mythology, urban legends do not replace the original story, they are, in a sense, competing with it. Having said that, urban legends are similar to mythology in that they will often lack the irrelevant details that we find in veridical accounts. Urban legends are trying to make a point, and so simply ignore the details that don't play a role in this. In my post on this, I argue that the people who originated an urban legend either a) simply made it up (i.e. they lied); b) hallucinated; c) experienced something they mistook for something else (such as nondescript lights in the sky which are mistaken for alien spacecraft); or d) were insane (didn't really experience anything, but now actually think they did). The biblical accounts of Jesus cannot fit into any of these categories. Rather than rehearse them here, I'll just commend you to my earlier post.
Incidentally, if you haven't read The Brendan Voyage, I strongly recommend it.
(cross-posted at Quodlibeta)