The following is an explanation given by Sean the software developer.
With "Dropbox Connection" Moneydance uses the Dropbox API to talk to the Dropbox service directly. So, you don't need to have Dropbox actually installed on your computer, but you do need to be online when Moneydance is running in order for changes to be synced. With "Drobox Folder" Moneydance will just read and write files to your Dropbox folder on your computer, which means that you need to have Dropbox installed and running, but you don't need to be online in order for Moneydance to read and write the changes.
If you won’t to change the sync method on your main computer set syncing to off. Shut down moneydance on any other computers. Disconnect any mobile devices. Delete the .moneydancesync file in Dropbox. Then follow the syncing procedure from the start using these articles. See http://help.infinitekind.com/kb/syncing-and-sharing-data