Essentially, it's a conversation by interested parties. It exists more in contrast to the bland, definitive, "this is how it is" DocumentMode. Indeed, a threaded conversation shows many viewpoints, shows the arguments for the viewpoints, lends energy to the viewpoints, and is typically the fastest way to elicit all the viewpoints. ThreadMode also adds an important sense of community. As someone commented in the early days of MeatballWiki,
ThreadMode is also easier to write because each contributor only writes a paragraph or two and stamps their signature afterwards (or not; anonymous contributions are also very frequent). Because the ease makes writing like this less affected, ThreadMode can be more fluid.
However, ThreadMode can quickly turn into ThreadMess. Because signatures are often redundant, and conversational segues and rhetorical blustering are unnecessary, threads have much more noise and fluff than DocumentMode. This makes them harder to refactor, as removing the essence of the discussion involves careful sifting of semantic content. Not to mention the somewhat random placement of interesting facts.
Furthermore, ThreadMode seems to work best when individual contributions are short and limited in scope to a single subtopic. Longer segments which touch on several related topics can lead to a multi-threaded mode, in which people respond to multiple parts of a contribution, eventually creating multiple threads which all progress simultaneously, possibly erupting into a ForestFire. This can be hard to assimilate and makes the abovementioned problems in refactoring worse.
Now, a wiki is freeform. You can either write documents or threads. Some media enforce threaded conversation, such as WebLogs. Actually, WebLogs traditionally place the article text at the top, followed by the threaded conversation. This shows how it is possible to combine documents and threads together, except on a wiki, anyone can just fix the document instead of blathering on about how it is wrong.
Note that it is not really possible to enforce document mode technologically as conversation will naturally erupt in some way. Check out the AdvoGato diaries (Advogato:recentlog.html) to see how this happens.
An interesting alternative would be DoubleWiki, which could act as debate and keep the pages (pretty) clean.
By the way, another, less important reason that threads are hard to read on wikis is that it is hard to skim the page and find where one comment ends and the next begins. That's why I often sign my name on its own line when I comment, rather than signing at the end of a line.