A major problem with a lot of the better fan fiction out there is dialogue.
See, the thing is, the better stuff has the broad strokes down. The writers understand the characters enough that if they have them change, it's for a reason other than the-story-wouldn't-make-sense-without-the-change thinking. There's a plot, of some sort, or if there isn't the character study is interesting enough that plot doesn't matter.
Then, there's the dialogue. It tends to be stiff and unnatural sounding. It's like the words in the characters' mouths are how the writers think people should speak rather than how people speak.
One of the big things they do is drop contractions. Yes, Data from Star Trek: TNG doesn't (can't?) use contractions, but everyone else on the ship does. People in life tend to drop their contractions for two reasons: 1. It extends the time you can think about what your saying allowing for a better constructed argument. And 2. dropping contractions helps to put emphasis on certain words and makes meaning stronger. English, as it's spoken, is a language that gets slurred, a lot, it's probably where contractions come from in the first place.
The other thing that really jumps out at me is the awkward word choices, like having character saying "fled" instead of "left" or other words or phrases that aren't used much in regular life. If one character chooses to use more archaic (for lack of a better word) or Latinate words, that helps to define a character. Awkwardly phrased dialogue can be used to show a flustered character, if it's used in the proper context. If all the characters speak with that kind of language, it becomes cumbersome and boring and sounds totally false. (Even worse is when it's only the characters using that kind of language and the narration is written very conversational.)
I think that the reason those writers write such poor dialogue is because they aren't listening to what they're writing. I know, that sounds crazy, but there's music in spoken language and, in my opinion, writers need to try and transfer that music to the page so that as people read it they can hear the music in their head. Like music, spoken language changes over time, what sounded natural 100 years ago doesn't necessarily sound natural today. I wish they'd listen to the dialogue they write rather than just read it.
Part of the problem is that I only really notice it when the rest of the writing is strong. Like I wrote above, these writers are decent with a strong sense of character and a reasonable sense of plot. Also, they have basic grammar down. There's the occasional mix-up with homonyms (I'm sure I fucked up some theirs and theres and they'res in this post.), but everything else is strong. And that strength is what makes the stilted dialogue stand out, at least for me.
Since it is fan fiction, the better writers are going to have lots of stories, which is nice. The problem is that these writers, at least so far, don't seem to improve. You can read a hundred thousand words worth of stories and not be able to tell which ones were written earlier and which ones were written later. I'm not sure if these writers don't want to improve or what. It's a shame.