something is wrong with your MySQL server (obviously lol).
check a couple things.
load a script with the following in it:
that will display your PHP version information, including PHP plugins. make sure MySQL is located somewhere in there (should have its own menu). if it is not in there, contact godaddy and let them know there is something wrong and either your MySQL server is not loaded, or the PHP extensions need to be reloaded.
BUT, based on the error there, it does find your MySQL server, but it says the authentication method is not supported.
what i think is happening is that vbulletin is trying to use the old PASSWORD() function in MySQL, and your MySQL version is using the new PASSWORD() function (they updated it between versions, dont ask why. my guess is security, i think it's a PITA).
do you have the newest version of vbulletin?