Unfortunately this isn't possible and I would suspect that it may never be available.
After doing a bit of research, I found a feature request for delta-syncing to be integrated into the API[*]. Dropbox hasn't responded, nor has the community upvoted this request.
I would make an educated guess that the reason why Dropbox hasn't provided this functionality, and likely never will, is because this is a dangerous feature in the hands of unknown developers.
Consider the case where you write an application that uses such a delta-change update system for updating large files. You thoroughly test your app and publish it to an app store. A couple of weeks after your initial release, and numerous downloads, you start receiving bad reviews and complaints because you managed to miss a very specific test case.
Within this specific, buggy case you've miscalculated a differential offset by 1-byte. Oh no! You've now corrupted thousands of files, for hundreds of users!
Considering such a possibility, I think I would personally request that Dropbox NEVER provide such a dev feature. If they integrated such a function into the API, they would be breaking their #1 purpose-- to provide consistent, safe, & reliable cloud backups of your important files.
[*]: This was the original reference location, but it is now a dead link.
(https://www.dropbox.com/votebox/1515/delta-sync-api-for-mobile-applications)