We think of Dropbox as a service for synching our directories, but the real value they bring is in applying a level of thoughtfulness that no one really applied to files before. A lot of that is part and parcel with storing this stuff in the cloud, which affords many user benefits—including availability of one’s files to countless third-party apps. But a lot of it is very particular to Dropbox’s superb design of the user experience.
I agree with Khoi Vinh's assessment of Dropbox's strengths in the era of apps and hidden filesystems. My work depends on Dropbox: all my text files are on it (through Editorial); it's the fastest way to share images across devices (I can't get AirDrop to work reliably between my iPhone and iPad most of the time); and, it's the backbone of the apps I use every day to publish articles and organize my research. I could work without Dropbox and use something like OneDrive or iCloud, but my workflow would considerably suffer. I'd be slower and live with the constant fear of losing control over files or, worse, the files themselves.
I also agree with the comment on the design of Dropbox. Features like versions, shareable links, and the recent addition of comments and recent files are all powered by a tasteful design that hides complexity and makes everything seem easy and seamless. I hope Dropbox continues to remain relevant.