The factions are very important for the setting and the plot and the characters; but they're also just big, big symbol factories. Almost everything about the factions is pretty symbolic of how their members think about the world. Take something super simple, something you have on you right now (we hope): clothes. In the real world, clothing may tell us something about a person, but Divergent takes this to the extreme, where every factions' clothing tells us how they see the world—and how they want the world to treat them.
Abnegation dresses all in gray so they don't stand out; and Tris and her mom "picked up nearly identical stacks of clothing every six months or so. It's easy to allocate resources when everyone gets the same thing" (8.100). So Abnegation clothes are about equality and blending in—which is what Abnegation is all about.
Amity dresses in red or yellow clothes and always comfortably (5.25). Because if your main goal is being happy and friendly, you want to look bright and feel fine. (Wear some too-tight shoes and try to be nice—you'll be yelling at puppies in no time.)
Candor dress in black and white because, as Tris notes, they see "the truth as black and white, so that is what they wear" (1.18). So the clothes-as-symbols aren't just something subtle that we're supposed to notice as readers—these are symbols even in Tris's world.
Dauntless dress all in black and are into piercings and tattoos (1.39). Tris recognizes that there's no clear connection between wearing black and being brave. But we recognize that the train-hopping, gun-wielding, Fight Clubbing. Dauntless aren't just brave. They're supposed to seem cool and rebellious and individualistic. If there were motorcycles in the future, Dauntless would be riding them because they seem cool. (And if there were factions in the 50s, James Dean would have been Dauntless.) The Dauntless with their piercing and tattoos are trying to stick out—the opposite of the Abnegation with their desire to blend in.
Lastly, Erudite wear some blue, but they have a very serious, knowledgeable reason for that decision. According to the Erudite, "blue causes the body to release calming chemicals, and 'a calm mind is a clear mind'" (28.14). So the Erudite all dress with at least some blue in order to have this effect. Candor people don't become more truth-telling because they dress in black and white; but Erudite are supposed to become more clear-minded because they wear blue. Just like Abnegation folks are supposed to blend in more because they all wear gray.
But here's one last question about symbolic clothing you're probably asking: why clothing in the first place? Why not focus on their modes of transportation or their initiation rites? Well, we think clothes are very important because (a) the book talks about clothes more than just about any other differences between the factions; and (b) because clothes are both social and individual. Whoa, wait, what?
If you think about it, Tris wakes in the morning and puts on her clothes. Those are her private, individual clothes, and she, personally, makes that choice. But at the same time, Tris gets her clothing from the faction she lives in. So, when Tris is in Abnegation, she wears her Abnegation clothes; when she's in Dauntless, she wears her Dauntless clothes. So we think of clothing as a personal choice (unless you have older siblings and get hand-me-downs, like we did), but that personal choice is tightly connected to one's position in larger society.