We strongly recommend that you use strictures (use strict; use warnings;) at the start of all the Perl you write. They give you early warning of a wide range of issues.

We also like it if you tell us up front if you are doing homework or are doing this as a learning exercise (this smells like a homework question) so that we can tailor our replies to help you learn.

Your first attempt is almost ok for small files, although the syntax is not correct and there is a problem. A syntax corrected version would look somewhat like:

use strict;
use warnings;

my $F1 = <<F1;
name   value
n1       001
n2       002
n3       003
n4       004
F1
my $F2 = <<F2;
name     value
n5       005
n2       002
n3       003
n8       008
F2

open FH1, '<', \$F1 or die "Failed to open F1: $!";
open FH2, '<', \$F2 or die "Failed to open F2: $!";

my @f1= <FH1>;
my @f2= <FH2>;

for my $f1Line (@f1) {
    for my $f2Line (@f2) {
        if ($f1Line eq $f2Line) {
            print "Same: $f1Line";
        } else {
            print "Diff: $f1Line";
        }
    }
}

close(FH1);
close(FH2);
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which prints:

Diff: name   value
Diff: name   value
Diff: name   value
Diff: name   value
Diff: name   value
Diff: n1       001
Diff: n1       001
Diff: n1       001
Diff: n1       001
Diff: n1       001
Diff: n2       002
Diff: n2       002
Same: n2       002
Diff: n2       002
Diff: n2       002
Diff: n3       003
Diff: n3       003
Diff: n3       003
Same: n3       003
Diff: n3       003
Diff: n4       004
Diff: n4       004
Diff: n4       004
Diff: n4       004
Diff: n4       004
[download]

which is not quite what you want and becomes horribly slow when the files get big.

Generally however this is a tricky problem to solve because good solutions require knowledge of the size and nature of the contents of the files. A common key to the solution involves using hashes (see perldata) to keep information about data that has been seen before.

At this point you can either play with your current code, or learn about hashes and possibly come back for more instruction.