... is that it breaks Wikipedia's internal mechanisms by totally and utterly destroying all sense of location in a discussion. In a Wikipedia discussion, each comment has a certain environment that may change, but usually not too drastically. Different discussion pages tend to have different visual flair: Large blocks of texts or lots of short comments, most comments indented on the same level or it keeps changing. 'Hatted' threads and sub-threads (i.e. you have to click to see them). Also, users can freely edit other users' comments.
There are advantages and disadvantages to the traditional wiki discussion style. Advantages include that you can usually see at a glance in which environment a certain comment that put a user into trouble was made, and if you want to be absolutely sure you can go back in the page history. And, very importantly, if you revisit a discussion after months, the visual appearance gives you clues that make it easier to remember what it was about and how it went and maybe even what you were going to say when you got distracted.
The very point of Liquid Threads is to move things around in such a way as to destroy all of that. Places such as Reddit and Stack Exchange have shown that this can work very well. But once you have a big audience and community norms built on a radically different system, I think it's problematic to make that kind of change. When I was still active on Wikipedia, Liquid Threads was already running on some meta-site. I felt that it was absolutely horrible to use because the re-ordering got in the way of exactly the kind of thoughtful discussion which that particular wiki was supposed to be for.
--- In case anyone wonders: After about 26,000 edits, I left Wikipedia in disgust for a year when it became clear that a vast majority of editors supported retaliating against Islamic extremism by angering ordinary, peaceful Muslims on the Muhammad article for no encyclopedic reason. (I would understand one or two Islamic Muhammad depictions to illustrate the fact that they exist - e.g. there is precisely one on the Turkish version of the article -, but half a dozen is way over the top, gives a very misleading - hence 'unencyclopedic' - impression, and seems designed exclusively to alienate Muslim readers and editors. This feeds the inferiority complex that causes some Muslims to become fundamentalists. By the way, I am an atheist and personally consider the Muhammad image ban stupid.)
When I returned I found that for whatever reason my ability to get *anything* done in controversial areas was gone completely. Apparently, using words such as "genital mutilation" in a discussion, applied to a gender for which the media of a large Western nation practising it on a large scale generally doesn't use it, is much worse than actually encouraging it in an article by abusing rules and then simply shutting down all discussion. And so I joined the ranks of ex-editors who complain about abuse by Wikipedia's almost completely uncontrolled admin caste.
Maybe Liquid Threads would even be capable of solving such problems, once it works properly and the community has adapted to it. Its introduction would no doubt cause a severe crisis, which, come to think of it, is probably just what the English Wikipedia needs.