Hm ... Maybe there is some kind of error on the attempt to store the download numbers in to one of these db table. If there is an error then it is logged in the error log of the server.
Which stat method are you using? "Use WP permalinks"?
You could also look how the query string looks like resp. whether the little redirect for the podPress statistic works or whether it is problem to write to these two db table.
Please, use Printphpnotices plugin (the one I have mentioned in this post) to log the strings of the db queries during the download counting.
For "Counts Only" insert the printphpnotices_var_dump call in line 915 and 917 of the podpress_functions.php file:
$sqlIoU = "INSERT INTO ".$wpdb->prefix."podpress_statcounts (postID, media, $method) VALUES ($postID, '$media', 1) ON DUPLICATE KEY UPDATE $method = $method+1, total = total+1";
$result = $wpdb->query($sqlIoU);
and for "Full" in line 944 and 946 of the same file.
$result = $wpdb->query($query);
If the printphpnotices_log.dat file contains a line like this
'INSERT INTO wp_podpress_statcounts (postID, media, play) VALUES (51, \'hallo.mp3\', 1) ON DUPLICATE KEY UPDATE play = play+1, total = total+1' after you have clicked on the player of one of the episodes then you will know that all works fine until the db action.
If this string is followed by a false then something went wrong during the attempt to execute this db query. If it is followed by a 1 or a 2 then the statistic tables should contain non-zero values.
The (1) behind the feed caching folder name on the general settings page of podPress is a good sign. It means that this folder is writeable for the script. Until 8.8.4 this test printed only a message in case of an problem with that folder. The "(1)" is some kind of a debug information but I guess only for me. I will probably hide this again in 8.8.6.
The "Podcast Feed URL" should be the URL which you want to (or would) submit to the iTunes podcast repository. No, I don't that using a feedburner feed address there can cause the problems with stats.
I'm hoping that the logs will bring us more information.