I don't understand this. You shouldn't need to rely 3rd party tools as phpbb has database backup built into it via the Admin Control Panel.
Oh yes, no problem with obtaining the backup. But I had understood from the results of my forum search that this was *not* the correct way to go about moving forum data between hosts.
You originally said you needed to get your forum db. Look in your config.php file and find the db name, then ask your other hosts support staff do this:
Lol - we can forget anything that requires help from a human being that end

Why do you think I moved?

Download the forum_backup.sql and restore it either by having your new host import it for you, or via Admin Control Panel db restore in phpbb, or via phpmyadmin db import, or the ftp & admin_db_utilities.php edit method described in the phpbb.com knowledge base.
Can I just recap: You suggest I obtain a "full backup" (ie structure and data) from my existing forum. I should then copy the phpbb2 directory from my old host over the top of my new one, preserving config.php as a guide to editing the one I copied over?
I think this is too difficult for me to get right. Maybe if I give the backup to the staff here they could install it for me?