In this snippet, you actually download the whole file each time you (cron) run(s) the script.

Not true.

From LWP::UserAgent, that LWP::Simple uses under the hood:

$ua->mirror( $url, $filename ) This method will get the document identified by $url and store it in file called $filename. If the file already exists, then the request will contain an "If-Modified-Since" header matching the modification time of the file. If the document on the server has not changed since this time, then nothing happens. If the document has been updated, it will be downloaded again. The modification time of the file will be forced to match that of the server.