You get most of the features of word, and the math looks better. And it's platform-agnostic! It doesn't matter whether you make that document on Windows, Mac or Linux, and it doesn't matter which software you use either. And cross-linking in the document is easy and a file-based database can be used to store the bibliography.
Word is very hard to you use once you get used to that system...it's too inflexible and a complicated document gets mangled when crossing the OS boundary.
That's only convenient for an already existing research group. The usual workflow of incidental cooperation, like a course instructor sending additional materials to students, is to generate a pdf from the LaTeX documents they wrote or otherwise obtained and e-mail it.
Matching citation styles used to be a problem also, but tools like Jurabib and Biber have mostly solved that problem.
Even more telling is MS stopped Office Live which didn't edit online in favor of Office 365 which does edit online. It's just a matter of time before it switches and the desktop version is an add-on while the cloud version is standard.
The GSA itself is a Google Apps customer. NOAA also made news this year by moving over its 25,000 users.
Then again, there are quite a few big corporations on Google Apps.
I've worked in research for several years and we almost exclusively kept LaTeX documents in a Git repository. This worked perfectly: Multiple people can edit the document at the same time and you can also add additional comments to the document that don't show up in the output.
Yeah, Google made a mistake in changing the default to Paginated. Still, Docs is light years ahead of Word + trying to emulate git's CVS via file names and mental notes.
I do however not have to do the edit-email-edit thing. We use SharePoint at work, which lets all of us make changes to a document in realtime.
This works inside and outside the company. The partners we typically collaborate with authenticate using identity federation. It works really well for us.
Use version control, or wikis.