I too would like to hear about how org-mode is an effective task manager. On its face, it seems like grandparent is saying "This doesn't compare with the power of Markdown for managing to-dos." One is a tool, the other is a formatting language.
If you're implying that TaskWarrior is a tool and that org-mode is a formatting language (or file format), then you're way off to begin with. Org-mode does have a specific format for its files. But much of its power derives from the tools it incorporates. E.g., the org-mode Agenda functionality enables complex searches over multiple files and collects and orders the results in a single buffer, and allows users to perform operations on the results (i.e., on the original buffers containing the results) directly from the Agenda buffer. There's much, much, more. Org-mode is primarily a tool, not a file format. If you want an overview of some of its functionality, you can take a look here:
http://doc.norang.ca/org-mode.html
Org-mode has embraced the emacs philosophy with respect to workflow management. It is a document formatting language which has been co-opted into doing pretty much all aspects of workflow management -- tasks, reminders, calendars, etc. I personally don't think it does a very good job, but some people swear by it.
I posted above why I like Org-Mode. Here are a few things it doesn't do well.
1. Because it is text-based, its visual summaries are limited. Like, there's a calendar, and for org-habits a simple color-coded chart to show consistency, but that's about it. If you're looking a week or a month ahead to visually see how many tasks you have coming up, the org-agenda is text-based.
2. While org-mode can sync with Trello, Beeminder, and a couple other task-management systems, its integrations are pretty limited. If you want to integrate with Contacts or whatever you'll have to work a bit.
3. org-mode's part of Emacs, so you'll need to be comfortable with Emacs.
Contrast this with Sunrise Calendar. Sunrise Calendar grabs gmail messages and appointments, iCal, google calendar, and displays them all nicely.
Pluses and minuses, depends on your needs and what you like.
