How does this service differ from Dropbox?

The services are similar to users, but Box has a lot of security integrations & features that tend to make it more IT-friendly. Depends how you want it to fit in with your organization.

DropBox is great when it comes to its functions and features and it's great if you want to use it to host more than just your documents (people can also host websites on Dropbox), while Box is ideal for permissions as it has very tight and comprehensive levels of access. Try using both and familiarise yourself with them to see which one works better for you.

Personally I prefer dropbox having used both of them, however I know some people who swear by box. As Bernardo says Box does offer you more in the terms of permissions, etc...

They are very similar. Primarily, I would say the storage offered is lower on Box, and DropBox has better UI and features.