Errors while running cron jobs often generate "mail" whether you've set it up or not.
You can likely check your mail by just running the mail command.

Also, in your crontab, add >> /tmp/cron.log after your script, at least you'll see whatever logs get generated from that.

Most likely though, the script works when ran manually because it uses your user's env variables, such as PATH.
I usually just use absolute paths for everything when using cron and this takes care of the vast majority of my issues.

Could it be a permission issue? Maybe the cron script isn't able to access the directory your calibre library is in.

Check your permissions and your working directory. Forgetting to make sure cron executes the script at the correct location is an easy and common mistake to make.

Theres no other logging it does? Have you tried running the script yourself from the commandline with the same privileges that cron has?