I'm not sure what the question really is, but maybe this will give you some ideas:

#Don't forget to use these!
use warnings;
use strict;

my %in_file_b;
my @aonly;
my ($file_a, $file_b) = ('Names_Ticker.txt', 'Names_Offer.txt');

print("Importing two data files \n");

#Look at FILE_B first since we will be eliminating elements 
#from FILE_A based upon its contents
open(FILE_B, "<$file_b")
        or die "Couldn't open $file_b for reading: $!\n";
while (<FILE_B>) {
    chomp;
    #the key will be the first four chars of the line, lowercase
    my $entry = lc( substr $_, 0, 4); #substr $_, -4 would get last fo
+ur
    $in_file_b{$entry} = 1;
}
close FILE_B;

open(FILE_A, "<$file_a")
        or die "Couldn't open $file_a for reading: $!\n";
while (<FILE_A>) {
    chomp;
    my $entry = lc( substr $_, 0, 4);
    #add the entire line if the value of $entry isn't in %in_file_b
    push @aonly, $_ unless exists $in_file_b{$entry};
}
close FILE_A;

print "@aonly\n";

1;
[download]

Processing it like this avoids slurping one of the files into memory.