Dropbox installs like a drive or folder on your computer, so you can save directly to it from your local computer as if it were a regular drive. You can also save any kind of file. To open or edit docs, you will still need to have a program from it (locally installed MS Office for example). But you don’t have to open a browser to use it, and there’s no additional downloading to your local computer to edit it locally, it’s just there.
Google Docs is useful for editing specific documents in real time with others, but not particularly useful for sharing multiple files. Both are cloud based, but have very different uses.
Dropbox gives you 2GB free, but if you get a referral, you get an extra 500MB, and the referrer gets an extra 500MB as well. So here’s my shameless attempt to get more space: http://db.tt/T2DsT63h