Box.net, unlike DropBox, more persistently targets business customers. This service has tough one precautionary features, access control and version control tools. The essential file types in business - are office documents, this is exactly why Box.net allows to view and edit them online (using Zoho editors). Box.net provides API for developers knowning that allows to integrate it to business applications. Recently Box.net developers added mainstream social tools - wiki, microblogging and profiles. According to the company info, Box.net has 50000 business-customers worldwide.
DropBox, similar to Google services, is closer to consumer market. (BTW it's interesting that Google owns domain dropbox.com). That's why they be more conscious of interface usability, easy-of-use and features, especially to produce sync between multiple computers. Its main advantage over Box.net would be the desktop client which allows to utilize files offline. But the service lacks sufficient user and version control. During September DropBox totally redesigned the web interface, improved the upload speed and unveiled the iPhone app (mobile access is another strength of Box.net). Recently this company owners announced they reached 2 millions users. One good reason for these a fast growth could be the free 2Gb subscription plan that DropBox provides to any or all.