You can solve it by running the following in the mysql rdbms.
SET PASSWORD FOR user@localhost = OLD_PASSWORD('password');
But since you are on a shared host, you should contact godaddy.com's support and have them fix the issue for you. The issue is that the mysql client's authentication protocol is not compatible with the rdbms server's authentication protocol in that particular machine. The script above simply tells the rdbms to treat your password (the one set by through the mysql client with the older authentication protocol) using backward compatibility by means of the older, deprecated authentication protocol.
I came across a discussion on the issue. Users said godaddy.com staff advice them to switch from windows to linux, and that this solved their problems. I'm not sure why godaddy would take this approach to fix a configuration issue, but I like linux better so, I'm not opposed to the initiative 🙂