At least one thing: Dropbox synchronizes its directory, meaning that every machine linked to your account has a local copy of the data stored there. Zumodrive keeps most things remote, unless you explicitly ask to have a local copy (seems to support some form of policies for which things should be local). In theory, then, Zumodrive lets you go beyond your disk capacity, but it's a bit too expensive to just offload hundreds of gigabytes.

Also, Zumodrive doesn't seem to have sharing like Dropbox does..?