I'm familiar with some of Armstrong's other works: "A History of God" and "The Great Transformation". I also realize she writes for a general audience so I was not expecting the discussion to be particularly scholarly, but I was expecting more than I got.
Now, it may bear mentioning that prior to Armstrong's turn to religious books, she was a Catholic Nun. I bring this up because her interpretation of everything which follows is clearly coloured by a tendency among western authors of religion to speak of religion in monotheistic terms. I've come to terms with the fact that YHWH gets top billing in such discussions, but when one considered the scope of the book, it becomes rather disappointing and problematic to her overall narrative. She almost entirely skips polytheism (arguably it could be counted among her discussion of neolithic animism, and in a brief discussion of different Canaanite deities), to the point of when speaking about the religions of ancient Greece, she mentions the proto-monotheism of Plato and Aristotle and little else. Considering the profound significance of polytheism in ancient religion, it is a huge gap to cross, and presents a major flaw in her work.
Secondly her idea of what a myth is, "...myth is make believe; it is a game that transfigures our fragmented, tragic world, and helps us to glimpse new possibilities by asking 'what if?' is not only terribly unhelpful, but I think considerably pejorative. When I think of myth, I think of a narrative structure, a framework for understanding the world, and giving it meaning; but I also think of it as true, if not always in an objective sense and here I find myself at variance with Armstrong, in the first two chapters she continually posits that "myth is make believe", that early humans knew that there really were no gods, but liked to pretend there were, to comfort themselves about their "terrible lot". Here again we see the Catholic worldview coming out in her observations; the world is a really awful, terrible place and humans will do anything they can to escape it, even concoct invisible friends to provide catharsis to the harsh realities of this wretched planet. Life is tough, life is a constant struggle, but for all that I think the world is a decent place and I have no desire to escape it. In a book whose central thesis rests on the merit and worth that myth offers to human cultures, Armstrong does as much to point out myth is nothing more than fantasy, but because it helps people, is still worth holding onto. I don't know, perhaps I'm just a theist set in my ways, but I happen to believe that the gods have an objective existence and are as real as you or I. Again, my bias I suppose.
Her framework for interpreting myth falls square into the Jungian/Campbell mold; all depiction of deity or hero is allegorical, and meant to be understood only in an allegorical sense. She establishes early on a dichotomy between allegory and literalism when it comes to understanding myths, and it is clear she believes it to be an either-or question. I prefer a more nuanced perspective (of course since I disagree with her definition of myth, I would); myth is not literal, because it is not history; myth is a narrative from which we impart meaning into the world and our actions within it, but this does not mean that all myth is allegorical either, because the myths are not necessarily describing anything other than themselves. Armstrong is also under the impression that myth is "dead" for all intents and purposes, and that it does not inform the modern, secular world view at all. This has more to do with how she defines myth (yet again showing her definition to be problematic), but the idea that myth is not still a way modern people in developed countries understand the world is at best naive and at worst symptomatic of Armstrong's own bias towards rationalism. One does not need to look far to see myth at work in any number of given subjects. J.M. Greer in his seminal "A World Full of Gods", provides an excellent example of this fallacy by speaking about the "myth of progress". One can see all sorts of other examples; the myth of America (aka manifest destiny) or the myth of linear time (as opposed to circular), each is a way in which people understand their world, but both are also mythic.
My other major problem with the book is that she seems to make a lot of conjecture and passes it off as fact. Most of the sources she cites in her chapters on the Paleolithic and Neolithic periods are considerably dated, and she makes rather sweeping generalizations. She also largely focuses on middle eastern cultures (with a few references to Hinduism, Taoism, Buddhism, Confucianism and some Greek), and wholly ignores any sort of Indo-European or P-I-E mention; to the point that when she speaks of early Greek myth, her interpretive framework comes from a middle eastern view point, which is simply untenable, knowing what we do of P-I-E mythology and how it informed the development of Hellenic religion and myth. To ignore such an important (especially to the western world) aspect of the developmental history of myth, in a book on the subject, shows the sort of tunnel vision Armstrong has.
Well those are the bad bits, how about some of the good? I do appreciate that Armstrong thinks myth has value, and is something which can have a positive impact on peoples lives. She adamantly refutes the idea of scriptural literalism and does a fair job of arguing some of the pitfalls of rationalism. I can certainly relate to her notion that myth can be used a a means of conveying important truths, especially when it comes to behaviour and ethics:
"The myth of the hero was not intended to provide us with icons to admire, but was designed to tap into the vein of heroism within ourselves. Myth must lead to imitation or participation, not passive contemplation."
Overall I'd say that the book isn't worth a read, for the small bits of useful or insightful information there is a lot of other opinion which provides little useful insight into the development of myth. Frankly there are far better books on the subject out there.