I use both Dropbox and Google Docs but I use them for different purposes.
I primarily use Google Docs to collaborate on “word” – style documents with my colleagues (other teachers). We use it to share lesson plans and work together to create assignments. The biggest benefit is that we can add information to the document at the same time. This is helpful when we have meetings together. Rather than one person being the note taker, all of us can contribute different parts at the same time.
There is also a chat feature you can use if you are all logged in to the document at the same time.
I tend to use Dropbox for two other reasons. The first is to keep any non “word” -style documents that I need to share with other people. Just last week, four of us (2 teachers and 2 admins) had to create one presentation. Myself and the other teacher taught the admins how to use Dropbox so we could share the PowerPoint and not worry about emailing the most recent version back and forth to each other.
A PowerPoint in Google Docs wouldn’t work as well because the formatting changes. However, in Dropbox, if two people are editing the file at the same time it will save two copies to the folder. This can be inconvenient.
The second thing I do with Dropbox is keep many of my personal files (mostly lesson plans) there because it works so easily between my work and home computers, my iPad and my phone. As someone mentioned above, it looks just like another window on your computer. The app for the iPhone and iPad is just as easy to use. (but there are issues with saving docs from iOS devices to Dropbox. It can be done, it just takes an extra step).
I am also using both of these with students with a reasonable amount of success. Using these tools with students is an ongoing process.
That being said, here is a shameless promotion of my own referral link – http://db.tt/qn5HhAs
This is probably more information than anyone really wants but if anyone is interested or has more questions feel free to PM me. I’d be happy to help.