I'm hoping someone will tell me why it's a security problem, but yeah, for a whole slew of apps, I'd like eliminate server logic or at least make it simple to swap out backends.
It is much cheaper to scale an application putting data onto peoples' Dropbox folders instead of S3. For S3, you would have to pay for the total amount of storage used by your customers; with Dropbox you wouldn't have to pay anything for storage, and for the most part your users wouldn't have to either since each gets a certain amount for free.
Exactly. With Dropbox, you're making storage the customer's problem and not yours. You could argue this is bad for the customer, but you end up with a web app with next-to-no server overhead: just serve your static files and everything else is taken care of. No different than the many mobile apps that sync to iCloud/Dropbox already.