Make sure your Backup account is set to use the same protocol as CuteFTP, such as SFTP or FTP (it won’t do both; pick one). I don’t know how CuteFTP does this transfer, but it doesn’t look like the Backup server will handle XFTP, which I don’t think CuteFTP has, either.
There’s really no point of backing up your site to the Backup account since your files are backed up via snapshot anyhow. However, if CuteFTP does the tedious download from one then automatically upload to the other, something should work. Can you use CuteFTP to do a regular upload to your Backup account?