You've not execution query.
$query = mysql_query($sql);
if(!$query){
die('Update failed!. Error:' . mysql_error());
}
This one is more clear format.
$sql = "UPDATE `my_db` SET `date_in` = '$today', `customer_name` ="$customer_name", `billing_address` = '$billing_address', `billing_suburb` = '$billing_suburb', `billing_city` = '$billing_city', `same_as` ='$same_as', `shipping_address` = '$shipping_address', `shipping_suburb` = '$shipping_suburb', `shipping_city` = '$shipping_city' WHERE `personID` = '$select'";
Hope this help.