This whole functions is not working correctly:

if (!$transfer->file_exists($phpbb_root_path, 'common.' . $phpEx))
{
$test_connection = 'ERR_WRONG_PATH_TO_PHPBB';
}

$phpbb_root_path is the global var for the directory of the phpbb forum, NOT the one you have filled in in the ftp form, which should be:
$transfer->root_path

if (!$transfer->file_exists($transfer->root_path, 'common.' . $phpEx))
{
$test_connection = 'ERR_WRONG_PATH_TO_PHPBB';
}

However, the function is still not working, not sure why not, but after I've removed it from the script, the installation just went fine .