It "just works" (if it actually worked).
With Dropbox, either you are storing everything for all of your users (and paying for it), or you integrate into the user's Dropbox account - which would necessitate a fancy ol' login/authentication system. Far from seamless, and relies on your users subscribing to something that is popular but far from ubiquitous.
So... either pay through the nose for all your users' storage, or force them to jump through some circus hoops on their own. Neither are great.
For apps that get a fair bit of off-line data creation this is a nice feature.
There's no way for a third party library to do this (at least not through iOS 6.1.3 SDK).
I haven't seen many complaints about iCloud's file syncing features, which are what Dropbox is the direct competitor to.
1) Google Drive - took my a long time before I even tried it, but it works about as flawless as dropbox for my needs. I actually like it a lot as I use Google Docs heavily, so it makes it easier to use those tools.
2) SkyDrive - offers the most base storage than the other. Has nice support for MS Office products.
3) AeroFS[1] - While it's still in beta, it looks promising (it has limited (android) to non-existant (iphone) support). [1] https://aerofs.com/