Just couple of small changes, and now both of your client and server are happy. I added my comments at each place I touched.

• Your main problem was in your client, when you checking whether it should end, you used =~, which shall really be !~;
• Couple of places you didn't check whether $buf is defined before you print, and I hate warnings, so added checking.
• At one place you did try to check whether $buf is defined, but you did:

  if (( defined $buf ) || ($buf =~ /\w/))
  [download]

  That logic was wrong, and it shall be:

  if (( defined $buf ) && ($buf =~ /\w/))
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  In your logic, even if the first clause results false (not defined), Perl will still evaluate the second condition. In my logic, if the first condition results false, the second condition will not be evaluated, and that's what you really want.

your server with my modification:
use IO::Socket;
use IO::Select; 
use strict;

my ($server, $client, $buf, $new_sock);
my ($SOCK_STREAM, $server_port); #pg: remove $buf, defined twice.

{

    my $listener=IO::Socket::INET->new( LocalPort => 21000, Listen => 
+2, Reuse => 2 );
        die "Can't create socket for listening: $!" unless $listener;
        print "Listening for connections on port 21000\n";
    
        my $readable=IO::Select->new;      
        $readable->add($listener);         

        while(1) 
        {
            my ($ready) = IO::Select->select($readable, undef, undef, 
+undef);
            foreach my $s (@$ready) 
            {   
                if($s == $listener) 
                {
                    $new_sock = $listener->accept;    
                    $readable->add($new_sock) if $new_sock;       
                    print $new_sock "Welcome!\r\n";                   
+                 
                } 
                else 
                {  
                    print $listener "Why don't you call me anymore?\n"
+;
                    
                    $buf = <$s>;   
                    print "$buf" if (defined($buf)); #pg: added if def
+ined
                    
                    if (( defined $buf ) && ($buf =~ /\w/)) #pg: it wa
+s ||
                    {          
                                if ($buf =~ /goodbye/i) 
                                {
                                    print $s "See you later!\n";
                                    $readable->remove($s);
                                    $s->close;
                                } 
                                else 
                                {
                                    print $s "You said: $buf\n";
                                }                  
    
                        } 
                        else 
                        { 
                        print $new_sock "Still there?\r\n";
                            $readable->remove($s);
                                $s->close;
                                print STDERR "Client Connection closed
+, because you said $buf\n" if (defined $buf); #pg: added checking for
+ defined $buf
                        }
                    }
        }
         }

}

your client with my modification:
use Socket;
use IO::Socket;
use strict;
my ($remote_host, $remote_port);
my ($internet_addr, $paddr);
my ($soc, $answer, $socket);

{
    use IO::Socket;
    
    $remote_host="localhost";
    $remote_port=21000;
    
    $socket = IO::Socket::INET->new(PeerAddr => $remote_host,
                                    PeerPort => $remote_port,
                                    Proto    => "tcp",
                                    Type     => SOCK_STREAM)
    or die "Couldn't connect to $remote_host:$remote_port : $@\n";

    
    do
    {
        print $socket "New host\n";    
        $answer=<$socket>;    
        print $answer;
    }while ($answer !~ /end/); #pg: was =~, it shall really be !~
    
    close($socket);
}
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