Dropbox seems to be more focused on teams of people working together. Dropbox is more focused on having a "shared folder". Kicksend seems to be more focused on individuals sending large files to other individuals. I could see Kicksend growing fast as the barrier-to-entry seems lower and particular use more focused, and in my experience, they are attacking a particularly more painful problem: sending large files to others.
From Drew's Dropbox YC application: "# What's new about what you're doing? * Most small teams..."
Also it's great how their desktop app acts more like an email-client for large files. It might not be as cool as having a shared folder in the cloud, but it seems like a superior path to adoption if the change to people's behavior builds on the way people do things today (attach files to emails).
Another quote from Drew's Dropbox YC application: "# What are people forced to do now because what you plan to make doesn't exist yet? * Email themselves attachments. Upload stuff to online storage sites or use online drives like Xdrive, which don't work on planes."