<?php
//File: /wp-content/themes/YOURTHEME/ index.php and/or single.php
//FIND:
if (have_posts()) : while (have_posts()) : the_post();

//ADD UNDER:
$twomonthsago = date("Y-m-d",mktime(0,0,0,date("m")-2,date("d"),date("Y")));
$twentydaysago = date("Y-m-d",mktime(0,0,0,date("m"),date("d")-20,date("Y")));
$currentdate = date("Y-m-d",mktime(0,0,0,date("m"),date("d"),date("Y")));
$postdate = the_date('Y-m-d','','',false);
?>

//ADD WHERE APPROPRIATE
<?php
if ($postdate > $twentydaysago) {
echo $postdate;
?>
//Insert HTML code here for data you want to show if the post is newer (within 20 days old of posting)
<?php } ?>

//notice the greater than/less than sign flipped
<?php
if ($postdate < $twentydaysago) {
echo $postdate;
?>
//Insert HTML code here for data you want to show older than 20 days ago.
<?php } else { ?>
//Insert HTML code here for data that is newer than 20 days.
<?php } ?>