the perl die command prints to STDERR. normally, if you want some text to be sent back to the web page by your CGI script, use print, even if its a failure message. So:

if ( error_condition ) {
    print error message to page;
    die error message to http server log;
}
[download]

(substitute appropriate text for the messages)
CGI::Carp writes no error to the error log, because the errors are sent to STDOUT instead of STDERR.
there may be flags to CGI::Carp to get the errors to STDERR also, but im no expert there...

UPDATE:
CGI::Carp does actually write to STDERR also, so that gets put in your error log too.