To be able to use a mask and not just get all files in the directory (which the above example does), I run a ls command on the directory, then runs the mask through a grep command and then gets those files.

So I modified your code below to do that (it is untested, but the premise should work fine for you)

sub ftp_start
{
    my ($ftp, $meetNo);
    my ($root, $filename);
    my (@directories);

    # the Perl regexp mask to get all files
    my $mask = ".";

    ### THESE 3 WORK FINE:
    $ftp = Net::FTP->new("atd77.acs.org", Debug => 0)
      or die "Cannot connect to some.host.name: $@";

    $ftp->login("USERNAME",'PASSWORD')
      or die "Cannot login ", $ftp->message;

    $ftp->cwd("$REMOTE_DIR")
      or die "Cannot change working directory ", $ftp->message;

    # Here is new code
    my @files = $ftp->ls("$REMOTE_DIR");
    @files = grep { $_ =~ /$mask/ } @files;
    for my $file ( @files ) {
        $ftp->get("$file") or die "get failed ", ftp->message;
    }

    $ftp->quit;
}
[download]

Updated to note that this version works with Perl RegExp as the mask.