Upon installation, root@localhost does not have a password.

You should be able to connect without a password by just doing this:

$ mysql -uroot

Try the following:

$ mysqladmin --no-defaults --port=3308 --user=root password 'somenewpassword'

or you can do it the stubborn way:

$ service mysql restart --skip-grant-tables --skip-networking
$ mysql -e"UPDATE mysql.user SET password=password('somenewpassword') WHERE user='root'"
$ service mysql restart
$ mysql -uroot -p

UPDATE 2013-01-02 16:47 EDT

I am sorry, I overlooked the port number.

Here is the problem: you cannot use root@localhost against a non-3306 port unless you use the TCP/IP protocol. Please try this:

$ mysqladmin --no-defaults --port=3308 --user=root --protocol=tcp password 'somenewpassword'

SHOW GRANTS only gives you back whatever you are connected as, which was root@localhost.

root@localhost and root@'%' and completely different users. Just do

SHOW GRANTS FOR root@localhost;
SHOW GRANTS FOR root@'%';

• root@localhost lets you connect from the DB Server via mysql.sock (the socket file)
• root@'%' lets you connect via TCP/IP, but you must explicitly connect with that protocol. Otherwise, mysqld connects you as root@localhost. This is true even if you attempted to connect using 127.0.0.1.

SUGGESTION #1

For the sake of security, you should use netblocks instead of '%'. For example, if you web servers at 10.1.2.20, 10.1.2.30, and 10.1.2.40, you should create

GRANT ALL PRIVILEGES ON *.* TO 'root'@'10.1.2.%'
IDENTIFIED BY PASSWORD '*fdfgdgdggfggfgfg' WITH GRANT OPTION;

instead of having a remote root.

SUGGESTION #2

If you really want the remote root, here is what you do

UPDATE mysql.user SET host='%'
WHERE user='root' AND host='localhost';
FLUSH PRIVILEGES;

That will do what you want, but remote root is not recommended.

SUGGESTION #3

Since root@'%' already exists, then remove the root@localhost.

DELETE FROM mysql.user WHERE user='root' AND host='localhost';
FLUSH PRIVILEGES;

This will leave root@'%' as the only root user, but you must connect explicitly with TCP/IP.