A big problem I see in your program is that you are reading the whole file into an array first and then are assigning that to a scalar so you have TWO copies of the whole file in memory. It would be more memory efficient just to read the file directly into a scalar.

use warnings;
use strict;

system 'clear';
print "Palindrome - gamma version\n";
print "--------------------------\n\n";
print 'Please enter DNA filename: ';
chomp( my $filename = <STDIN> );

open DNASEQ, $filename or die "Cannot open $filename: $!";

( my $dna = do { local $/; <DNASEQ> } ) =~ tr/atcgATCG//cd;

my $count_of_2 = () = $dna =~ /a(?=t)|t(?=a)|c(?=g)|g(?=c)/ig;

print "Number of 2bp palindromes: ", $count_of_2, "\n";

__END__
[download]

See if that produces the same result as your friend's C++ code.