I use Transmit for the Mac. Maybe Filezilla can do this also:
If I open an FTP (SFTP, actually) window to one site, and a new FTP window to another site, I can drag files from one FTP server window to the other FTP server window. It's a little cumbersome since it's routing all the files through my home computer, but it acts like a server to server transfer.
If you are handy with command line, you can log into your shell account here and FTP to your GoDaddy account and try a recursive FTP get: