This is untested, but could run roughly 200 times faster than your original. Also, from my reading of the POD, the output file should contain both sequences and their ids.

#!/usr/local/bin/perl
use strict;
use warnings;
use Errno;
use lib " /RemotePerl";
use lib " /System/Library/Perl/5.8.6";
use lib " /Library/Perl/5.8.6";
use Bio::Perl;
use Bio::SeqIO;
use Bio::SearchIO;

if (@ARGV < 2) { die "usage: compare.pl <filename1> <filename2>\n"; }

my $file1 = $ARGV[0];
my $file2 = $ARGV[1];

# Open first fasta file
my $File1 = Bio::SeqIO->new(-file => $file1, -format => 'fasta');
print "File name of the first fasta file is: ".$file1."\n";

#Create  a hash from file 1
my %fasta1;
while( my $seq1 = $File1->next_seq() ) {
    $fasta1{ $seq1->seq } = $seq1;
}

# Open second fasta file
my $File2 = Bio::SeqIO->new(-file => $file2, -format => 'fasta');
print "File name of the first fasta file is: ".$file2."\n";

#Setup output file: sequences from File2 which match sequences of File
+1
my $output = Bio::SeqIO->new( 
    -file => ">EQUAL_HITS.fasta", -format => "fasta", -flush  => 0
);

# write matching sequences of file2 to the output file
# Note that if several matches to one sequence exist, 
# ALL matches are output to the file

while( my $seq2 = $File2->next_seq() ) {
    if( exists $fasta1{ $seq2->seq } ) {
        $output->write_seq( $fasta1{ $seq2->seq } );
    }
}

print "Comparison is complete! \n";
[download]

The main change is that it creates a hash rather than an array from the sequences of file1. This makes the lookup O(1) rather than O(200).

It stores the sequence objects returned by next_seq() as the values, keyed by the sequence, and when matches are found, it give the sequence objects back to write_seq(), for inclusion in the output file. Hopefully, write_seq() knows what to do with them.

Other things to note: use strict is not commented out; I've shortened some of your variable names; the code is indented; as you've set up variables $file1 & $file2, you might as well use them rather than $ARGV{n].