MySQL - ODBC connect fails, Workbench connect works

Solved.

As it turns out, it was a permissions problem. I ran the following command on the remote server SQL:

GRANT ALL PRIVILEGES ON *.* TO 'root'@'(my_host)' IDENTIFIED BY '(my_password)';

I had run the previous command, but without the "IDENTIFIED BY" password. Then, to reset the mysql permissions cache, I also ran

FLUSH PRIVILEGES;

And now it works.