Neither the user used to run the task nor the user connecting to the other domain
Do both domains have an account with the same name? Is there a local account on the server with the same account name?
Again, I can run the script manually with no errors, but when I try to run it using task scheduler I get this error.
Check the security eventlog on the "source directory" server. You should see some logon event for that account or an event that references the machine that runs the script. Does it show the correct account domain?
Build a test script (bat file) to verify authentication to that one server.
net use x: \\server\share password /user:Domain\user
dir x:\
net use x /delete
Update: be sure to capture all stdout and stderr for any task that you run. In the task definition set it to execute cmd.exe. In the arguments field add the redirection. It should look like this:
cmd.exe /c c:\scripts\YourScriptName.bat 1> c:\temp\myscript.log 2>&1