Something like this:

# assuming you've got the files open in handles $fh1 and $fh2
while (1) {
    last unless defined($line_from_file_1 = <$fh1>);
    last unless defined($line_from_file_2 = <$fh2>);
    # ...
}
[download]

Basically you just use the "read a record" operator (<>, aka diamond operator) on each file handle. See also readline.

Phil

use warnings;
use strict;
open my $fh1, '<', 'file1.txt' or die "Can not open file1.txt $!\n";
open my $fh2, '<', 'file2.txt' or die "Can not open file2.txt $!\n";
while (<$fh1>) {
    chomp;
    my $line_f2 = <$fh2>;
    chomp $line_f2;
    if ($_ ne $line_f2) {
        print "Mismatch at line $.\n";
        print "file1: $_, file2: $line_f2\n";
    }
}
close $fh2;
close $fh1;
[download]

Your code silently ignores extra lines in file2.

Yes, I realize that, but thanks for explicitly pointing it out. The OP stated that the 2 files have the "same number of lines". So, I presumed that the OP would be checking that ahead of time.

  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;
[download]

By your hashbang line, it appears that you're on a unix box, so you might try:

paste file1 file2 >outfile

...roboticus