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Sharing and mobility is where Google Docs completely trounces OpenOffice. If you want to move your OpenOffice documents around and share them with other users, you'll need to use a USB drive, send them as an email attachment, or use another web-based file sharing solution like
Dropbox. Google Docs, however, has built-in support for sharing documents with other people with fine-grained permissions; you can allow them to only view your documents, or
modify them.
Google Docs is also much more mobile. You can log in to your Google account and have instant access to your documents from any location in the world with Internet access. Or, more practically, you can easily keep your documents in sync between a laptop and desktop computer without having to manage moving them around and remembering which is newer.
In fact, not only can you log in to your Google account from any computer, you can log in to it on smart phones such as the iPhone and Android-based systems and view and edit your documents on the go. OpenOffice doesn't run on smart phones, of course.
Winner: Google Docs