It could be one of a couple of things.
First, if you have MySQL version before 5.1.6, you'll need the SUPER permission no matter what:
http://wiki.civicrm.org/confluence/display/CRMDOC/CiviCRM+MySQL+Permission+Requirements
However, since it has been running fine and you're just trying to import the dump file, I think it's probably that your dump file has functions where the definer is set as the database user for the old site (let's say "foo@localhost"). You aren't allowed to set the definer as someone other than yourself without the SUPER privilege. If the new site has a different user (maybe "bar@localhost"), you're acting as "bar@localost" trying to set a function defined by "foo@localhost".
In your dump, you'll probably see something along the lines of
CREATE DEFINER=`foo`@`localhost` FUNCTION ...
You can do one of two things:
- have both databases use the same database user account, or
- edit the dump file to replace the old user name with the new user name everywhere it occurs.
You should also be aware that "foo@localhost" and "foo@%" are separate users, even if they have the same name and password.