this is more a css problem than wordpress related –
check style.css of your theme: the ‘border’ (actually a padding with background image) is only set for left, right, and center aligned images (like those from your first link), not for those with .alignnone (those from your second link):
#content img.alignleft,
#content img.alignright,
#content img.aligncenter {
background:url("images/wf-hashes-wht.gif") repeat scroll center center transparent;
margin-bottom:12px;
padding:6px;
}
if you want the same behaviour for non-aligned images, add #content img.alignnone to the above list (don’t forget to add the comma)