The key feature of TiddlyWiki is that instead of WikiName links leading to new pages, it displays a new set of microcontent at the top of the current page, pushing your previous content toward the bottom of the page. Thus your link history is all on a single page rather then through your browser's back button. You really need to try it out to understand it.
In its current form, TiddlyWiki is still really more a technology demonstration than a useful tool, as you can't save the changes you make. However, there are people that are creating a PHP TiddlyWink based on it.
There are a number of things about TiddlyWiki that interest me. One is that it is more microcontent focused, rather then page focused. I've seen a few attempts to do blogs in Wikis and to date I have been unsatisfied -- part of the reason is that most blogs consist primarily of a chronological series of microcontent, whereas Wikis tend to be more oriented around pages. The ideas in TiddlyWiki may offer an alternative.
Another unique aspect of TiddlyWiki is its approach to link history. I'm not sure if new microcontent should appear at the top, as it does in TiddlyWiki, or at the bottom, but it certainly is an useful experiment.
TiddlyWiki also seems to also be an interesting way to create hypertext stories -- see for instance see the interactive fiction Baby Dog Sitter.
I look forward to seeing what shakes out as this tool matures.