It can be as simple as this:

use Modern::Perl;
use IO::All -strict;
use List::MoreUtils qw /each_array/;

my @first_array = io('./first.txt')->chomp->slurp;
my @second_array= io('./second.txt')->chomp->slurp;

my $ea = each_array( @first_array, @second_array );
while ( my ( $first, $second ) = $ea->() ) {
    if ( $first ne $second ) {
        say 'Arrays differ at line ', $ea->('index');
        say "First:  $first";
        say "Second: $second";
        last;
    }
}
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It is simple and fast but not memory efficient as it reads in the whole files before starting to compare them.

This one is more memory efficient, but might be slower (depending on the length of the files and where the first difference is found):

use Modern::Perl;
use IO::All;

my $first_file  = io('./first.txt')->chomp  or die $!;
my $second_file = io('./second.txt')->chomp or die $!;

my $index = 0;
while ( my $first_line = $first_file->getline ) {
    my $second_line = $second_file->getline;
    if ( $first_line ne $second_line ) {
        say "Files differ at line $index";
        say "First:  $first_line";
        say "Second: $second_line";
        last;
    }
    $index++;
}
[download]

Update 1: added a more memory efficient version.

Update 2: used the slurp method in the first version.