Show only the grandchildren (using get_terms)

I think for each term you should check that it has a parent but not have any childern

So you code may look like

    $taxonomy = 'world';
    $tax_terms = get_terms( $taxonomy );
    foreach ($tax_terms as $value){
        $args=array(
            'child_of'=> $value->term_id,
            );
        //get all child of current term
        $child = get_terms( $taxonomy, $args );
        if( $value->parent != '0' && count($child) =='0'){
            echo $value->slug;
            echo '<br/>';
            //do something because it's your lowest level term which have parent but not have any childern
        }
    }