Dropbox synchronizes files between your computer and their servers. Keybase stores the files on their servers and only downloads/uploads them "on demand". Think of your web browser accessing gmail.com versus running something like Thunderbird that downloads your mail locally.

As a benefit, it's much simpler to implement - you don't need to work as hard to handle conflicts where two people made incompatible changes while offline. However, it will be slower for bulk operations and will require an internet connection.