Well it depends of what exactly you want to do, if you only want to move the table showing the additional attributes, you’re right it has to do with additional.phtml, just move in \app\design\frontend\default\default\template\catalog\product\view.phtml the following code where you want the table to be.
<?php if ($_additional = $this->getChildHtml('additional')):?>
<h4><?php echo $this->__('Additional Information') ?></h4>
<?php echo $_additional ?>
If you want to display only one specific attribute anywhere in the view.phtml page, you can directly call the value like that :
// if it's a text attribute called for example 'attribute_name'
// if it's a select attribute called for example 'attribute_select'
Refer to this page if you want to display your custom attributes in list.