I’ve been asked this question lately; Livedrive’s blog answers it nicely:
Backup keeps a copy of all of the files that are on your computer in the cloud. It’s very easy to use and install and you don’t have to change the way you work. Your files are just in the cloud, and you can view them on the web or from your phone, and restore them back to your computer at any time. However if you want to access these files from another computer you have to log into the web portal to view them – they don’t sync between your computers.
Briefcase is your own personal cloud storage space, and appears like a new drive on all of your computers. You can use the drive just like any other disk drive. Files you save to that drive are synced between all of your computers – and in a similar way to Backup you can access them from the web or your mobile devices. Briefcase makes it easy to work from anywhere on the same files – without emailing yourself or carrying around a USB stick. Briefcase also lets you share files with friends and family with one click.
Read their whole post to learn more.