iCloud is the native sync solution for Mac and iOS platforms. It syncs automatically in the background which makes it faster than Dropbox. It’s also included with your Apple account and is automatically enabled if you’ve setup iCloud Documents and Data on your device. Bonus: there’s no need to create a new account.
That being said, iCloud is still very new and we’re currently working to improve integration with Day One. For example, the initial load of large libraries using iCloud can make it seem like the app is frozen. We plan to make this an even more seamless experience very soon.
Dropbox is still an excellent alternative and also offers some benefits that are not available on iCloud. It’s a proven platform, but since it’s not a native solution, we must perform the syncs on specified intervals. It also requires you to create an account, which isn’t convent for new users.
Dropbox does offer a great backup solution in that all data posted to Dropbox is saved with a history of up to a month (in addition to the auto-backups created in the Day One Mac app). So even if you delete all your entries, they can be recovered on dropbox.com.
Both options offer a great advantage to not syncing at all. Even if you’re only using Day One on a single device, your data can be saved in the cloud so you can regain access if your device is lost or broken.
You can’t go wrong with either option.