Factual Story vs Mythological Story: Is One "Truer" than the Other?

 I have noticed a pattern with this class (and how fitting is that since the point of this class is to observe repetition) and it is that I am consistently having "light bulb" moments when we discuss the interconnectedness of, well, everything. I feel as though I spend most of my time in class thinking, "Ahhhhh, now I see it!" or "Oh that makes a lot more sense now." And while I love the learning process, I have been feeling that I am maybe not knowledgeable enough to be able to assert my view on elements we have been discussing. However, when "factual story vs mythological story" came up in class today I found myself twitching in my seat out of eagerness to discuss the topic.
The debate over fact vs myth is one of my favorites to engage in ever since I read Tim O'Brien's The Things Carried.  In his book, O'Brien recounts his experience in war through story, but a separate dialogue informs the reader that those stories don't always accurately retell the facts. While some readers may find this upsetting, I fell in love with the notion.
Why should we be restricted by facts in sharing our experiences? Some times the best way to explain what happened is to describe it in terms of how it felt or seemed or what it looked like from one's own perspective. After all, what we choose to do and how we choose to live our lives is based on our own perception, so we may as well be sharing that with others since it is, in that way, truer and more relevant.
In John Cheever's "The Swimmer," we see an old man journey across fences and through yards to swim through a series of pools to get home. He insists that he is swimming around the world. Now, Cheever could have described this story from a view point of the man being a crazy alcoholic that just swam around, but by allowing us to peak inside of Ned's (the man swimming) mind, we see that this is his reality and we are given the opportunity to experience it alongside him.
The idea of story truth being truer than facts appears in Sven Birkert's "The Art of Attention" as well.  Birkert writes, "If a thing doesn't necessarily matter in itself it might matter because of what it shows about something else." Take, for example, someone's recollection of a horrible day they had. Let's say they describe all the things that went wrong on that dark and rainy day. Does it really matter if it was, in fact, a bright sunny day? Or is it more important for that person to convey the essence of their experience by describing the day as dark and rainy because that is how it felt to them?
While facts are clearly necessary in life, I do not think they are the key element of a story. It seems to me that it is just as - if not more -  important for someone to convey their experience in as raw and personal of terms as they can than that they make sure they have the date and time correct.
You tell me which picture you can empathize with most.