my @Fn = qw' bigfile1.txt bigfile2.txt ';

  open my $f1, '<', $Fn[0] or die "$Fn[0] $!";
  open my $f2, '<', $Fn[1] or die "$Fn[1] $!";

  while( defined(my $l1 = <$f1>) &&  defined(my $l2 = <$f2>)) {
     # chomp here, but only if necessary
     print "line $. is ", $l1 cmp $l2 ? "different!" : "identical!", "\n"
  }
  close $f1, close $f2;