File::Copy calls croak when an error happens. The easiest way to prevent croak from outputting to STDERR is to wrap the call in an eval:

  eval {
    copy($fileOne,$fileTwo);
  };
 
  if( $@ ) {
    die "Did not copy: $@";
  }
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Update: then again ... since you've re-opened STDERR to a file and since croak calls die and die outs to STDERR ... there's really no need for any of this (unless your platform does something funky when re-opening STDERR).