At long last, all the words are on the page. You have (hopefully) run the spell-checker and had a friend look over it for obvious errors. Why shouldn't you post your newest, shiniest story right away?
That depends on why you're posting the story. If writing it was your only goal, you're done, and there's no need to ever post it to a public forum. But if you want something more from your writing efforts (recognition, shared pleasure, constructive criticism, community, whatever floats your boat), the time between completion and posting is your opportunity to increase your chance of getting it.
Writing and Storytelling
There are people who really love the writing. They love the craft, the metaphor, the structure, the style. Others really love the story, the image in their heads and their passion to get it on the page, and are less interested in "technique." The two are not mutually exclusive and, in many ways, can be complementary. Think of a spectrum, like the Kinsey Scale, with most people falling somewhere in the middle, but closer to one end or the other.
Motivated as they are by love of craft, it may be easier for writers to enjoy the process of polishing their story and really making it "finished."
I see the storytelling process as a sort of emotional loop:
Done vs. Finished
Getting the story down is only the first step in creating the finished piece. The first draft is not the finished story, no matter how good you are.
Many storytellers want to throw their first draft out to the masses. Many do. And some readers enjoy it enough to respond. What if, by taking a little extra time, you could expect more feedback? More interaction with other readers/writers? More community?
Getting What You Want
Now direct your attention back to the emotional loop. For the purpose of this article, I assume that you (the fanfic author who posts her story, rather than leaving it to linger on her hard disk) want your readers to have an emotional reaction to the story (and I don't mean that they hit the delete key and curse your name). I assume you'd like to hear what they thought of it, and maybe even have a discussion about the characters, their motivations, and possibly even spark ideas for your next story. I assume this will be fun and feel good to you.
My assertion is that writing well (as well as you're able now, and working to get better over time) will improve your likelihood of getting the sort of feedback you want, of making an emotional connection to your fellow fen, and generally improving your own storytelling experience. And it goes the other direction, too.
No amount of craft will insert passion that does not exist. Poor craft can completely obscure the purest passion.
The things you're writing about matter to you. If they didn't, you wouldn't have spent the time and effort you did writing the story. Every little mistake you make (and we all make them) has the opportunity to throw a reader out of the story. Bad comma placement? Suzy just hit the delete key. Poor characterization? You've just lost Sandy. Overuse or misuse of dialect (characteristic speech patterns)? Angela just moved on to the next story in the archive.
Details matter. Not for any overall "rightness" or a better grade in English composition, but because they can make or break the experience for many (I think most) of your readers.
But when you take the time to get it right, when you remove those trivial stumbling blocks, your readers have the very best chance of "getting" your vision of the characters and the story you've chosen to tell . And then you have the best chance of creating emotional resonance that will carry you into the next story, and the next, and new and rewarding relationships with your readers.
I've been around fanfiction for a while. A really long while. I'm almost out of fingers and toes, if I start counting the years. I've read a lot of fanfic. I've written a fair amount. I've edited more than a dozen fanzines. I guess I'm just trying to say, I think I know of what I speak. That said, of course everything here is my opinion, and may not apply specifically to you.
I have an additional (and separate) personal need to write as well as I can to avoid embarrassment. Given that many of my friends are writers of far more skill and experience than I have, I do my best to polish my prose to avoid looking like an idiot.
Art, noun, skill acquired by experience, study, or observation
Craft, noun, skill in planning, making or executing
Usage Note: Craft has been used as a verb since the Old English period and was used in Middle English to refer specifically to the artful construction of a text or discourse. In recent years, crafted, the past participle of craft, has enjoyed a vogue as a participle referring to well-wrought writing. Craft is more acceptable when applied to literary works than to other sorts of writing, and more acceptable as a participle than as a verb. Seventy-three percent of the Usage Panel accepts the phrase beautifully crafted prose. By contrast, only 35 percent accept the sentence The planners crafted their proposal so as to anticipate the objections of local businesses.