wow.
it sounds like the web server is trying to log in with a name/password that isn't right.
probably the new web server just uses a different name than the old one did.
can you determine what name it is trying to use?
The other possibility is that the login name is the same but the database permissions restrict where that user can connect from, and because you've moved to a different server you're coming in from a different address.
you can see the names and where they can login from, by querying the mysql database. use the database named "mysql", then "select Host,User from user" - this will show all users and where they connect from.