The code the OP provides is suggestive of an approach that works if the word list is sorted.

#!/usr/bin/perl

use strict;
use warnings;

my $num_duplicates = 0;
my $last           = '';
while (<DATA>) {
    chomp;
    if ( $_ eq $last ) {
        $num_duplicates++;
    }
    else {
        print "$_\n";
        $last = $_;
    }
}

print "There where $num_duplicates duplicates.\n";

__DATA__
apple
banana
banana
cherry
date
date
date
elderberry
[download]

Output:

apple
banana
cherry
date
elderberry
There where 3 duplicates.
[download]

Since the OP had the count variable, I kept it to track the total number of duplicates in the list.