Recently a couple of services offering online sotrage and a software client for easy synchronisation between several clients, easy sharing and backup came up. It seems like the idea is actually quite old, but as I think really sexy.
There is a nice video at DropBox's webpage explaining what those services are good for, so I won't explain these advantages anymore but redirect you to the video.
If I wanted to backup all my valuable data I would need more than 100 GB, which is more than the 50 to 100 GB both services offer to paying customers. But just for quickly sharing files or making some important files available everywhere or similar things the 2 GBs offered by both services are fine.
While UbuntuOne comes with an open source client which can be trusted, adapted and ported to other plattforms DropBox is a proprietary closed source product. Funnily enough DropBox is also available for Mac and Windows while UbuntuOne isn't, yet. The DropBox client needs some tricks to install on a Ubuntu and probably other Linuxes, too. But in the long run I see a bigger potential in UbuntuOne being ported to anything you could possibly wish for just because it's open source.
Both clients integrate very nicely with Nautilus and enable you via right click to share files or folders. Only details differ here.
Featurewise those clients are not very different either, but the way it works is a little different. There is only one feature I really miss at UbuntuOne and why I use Dropbox. With DropBox you can share files via a simple http link while at UbuntuOne you can only share files when both partners have a launchpad.net account. This is very bad for the very typical situation where you want to send a large file to someone. DropBox's direct link you can obtain by right clicking a file comes in very handy in this use case.
Just now I prefer DropBox over UbuntuOne because of the Windows client being available and not needing any client or account for sharing files via direct weblink. But I predict bigger potential for UbuntuOne because of better integration possibilities with Ubuntu, easier installation and it being open source so there can be clients for any OS and architecture.