Drawbacks: Requires Flash for end-user viewing. It's currently somewhat difficult to install, although a one-click install is scheduled for this summer. And because it's so powerful, some users say that implementations must consider how to winnow down functionality so as not to overwhelm end users.
Skill level: Intermediate for those just creating visualizations; Expert for those implementing a Weave system.
Learn more: The Weave site includes demos, videos and a user guide. For more examples of visualizations that can be built using a Weave platform, see one planner's MetroBoston DataCommon gallery. In addition, I wrote more detailed Computerworld coverage of Weave following a presentation at Northeastern University.
Related tools: Tableau Public is a robust general-purpose visualization platform.
Also see: 22 free tools for data visualization and analysis (April 20, 2011) and Chart and image gallery: 30 free tools for data visualization and analysis.