The main benefit of online word processors is their availability from any computer with an Internet connection. You can't count on being able to access the same software on one computer that I used on the last — and unfortunately, although most modern file formats can be read by any word processor, there's always the risk of losing formatting, pagination, or fonts opening a document created in one program (or version) in another.
Using an online word processor means to have a standard format and interface from computer to computer. You don't have to worry about whether the version of Word on this computer matches the one on the computer where you started your document, or whether you won't be able to open it at all. You just log in to Google Docs or Buzzword and continue where you left off.
Off-site storage and backup is another advantage of online word processors. No matter what kind of trouble you get into, your documents are still safe and sound on servers hundreds of miles away.
When it comes to collaboration collaboration/sharing, most online word processors beat even the mighty Word, hands down. Documents can be worked on live, rather than emailing copies back and forth and trying to keep track of versions. Plus, most online word processors allow you to set various levels of permissions, so that you can offer read-only access to one group of viewers, full editing privileges to another, and the ability to add comments to a third. You can often post documents directly to the Web, too, which can be quite handy.
Finally, some online word processors just have good user interfaces. Google Docs is simple, streamlined, perfect for just opening a document and slapping some thoughts together. Adobe's Buzzword, on the other hand, is simply gorgeous.
The main weakness could be security or ... the perception around letting someone else guard your precious texts and secrets. But, if you feel safe enough about leaving thousands of dollars in someone else's bank, why not do the same with your writings. However, hardware isn't inherently more secure than web-based solutions — look at how many laptops, hard drives, and thumb drives get lost by businesses and even national security agencies like the FBI!