Permission for specific role to edit ANY comment
The permission drupal gives by default says 'edit own comments'. I do not see a permission which allows 'edit any comment'. Try using User Tweaks module. Module page says


  edit any comments (and edit any  comments), which allows
  user with the Post comments permission to edit comments. The
  permission is not required for users with the Administer comments
  permission.


Updated Answer-

I just saw the comment module code.
You simply have to give Administer Comment Permission for that Role, but it will give delete permission as well.

If you are comfortable in custom coding you can alter the access callback for comment edit menu item.

If you see hook_menu implemented by comment module, here is the delete menu item code

 // Every other comment path uses %, but this one loads the comment directly,
  // so we don't end up loading it twice (in the page and access callback).
  $items ['comment/%comment/edit'] = array(
    'title' =&gt; 'Edit',
    'page callback' =&gt; 'comment_edit_page',
    'page arguments' =&gt; array(1),
    'access callback' =&gt; 'comment_access',
    'access arguments' =&gt; array('edit', 1),
    'type' =&gt; MENU_LOCAL_TASK,
    'weight' =&gt; 0,
  );


which basically calls comment_access function as 

function comment_access($op, $comment) {
  global $user;

  if ($op == 'edit') {
    return ($user-&gt;uid &amp;&amp; $user-&gt;uid == $comment-&gt;uid &amp;&amp; $comment-&gt;status == COMMENT_PUBLISHED &amp;&amp; user_access('edit own comments')) || user_access('administer comments');
  }
}


You can implement hook_menu_alter to change access callback for edit, something as below.

  // Implements hook_menu_alter
  function mymodule_name_menu_alter(&amp;$items)  {
    $items ['comment/%comment/edit']['access callback'] = '_my_module_name_custom_callbck';
  }


// Access Callbck for comment edit.

  function _my_module_name_custom_callbck($op, $comment) {
     if ($op == 'edit') {
     global $user;
      return (in_array('your_user_role', $user-&gt;roles) || $user-&gt;uid &amp;&amp; $user-&gt;uid == $comment-&gt;uid &amp;&amp; $comment-&gt;status == COMMENT_PUBLISHED &amp;&amp; user_access('edit own comments')) || user_access('administer comments');
    }
  }


Note: I have not evaluated why drupal is not giving it by default, it will still be important to identify.