As the other posters have remarked, Brent's sync diary provides some of the concrete technical reasons why we rolled our own.
Big picture, though, we consider sync so essential to Vesper's long-term success that we wanted to control it. Dropbox is fantastic. We all rely on it personally. But we don't want Vesper to rely on it. I'm sure everyone here on Hacker News has a Dropbox account, and I'll bet most Daring Fireball readers do too. But we want Vesper to appeal to everyone, including people who don't know what Dropbox is or don't want it. (Plus, Dropbox doesn't work exactly the way we want it to work; we have code running on our servers, not just data stored on them. Vesper Sync works exactly the way we want sync to work.)
iCloud is very tempting too, because it puts all the onus of identity, privacy, and security on Apple's shoulders. But with iCloud, you have to choose between document syncing and Core Data -- neither of which is a perfect fit for the way Vesper is designed.
Lastly, we want to keep the door open for a web app version of Vesper, and a non-Mac App Store version for OS X. iCloud, by design, only works for apps distributed through the App Store.
We wanted to offer one and only one good way to sync. That meant rolling our own.