I would have probably used a different approach and not relied on the modified header from the server. For tasks like these, I'm in love with MD5 ...

#!/usr/bin/perl -w 
 
use strict; 
use MD5; 
use LWP::Simple; 
 
my $url='http://www.mrnick.binary9.net'; 
my $file='sitechanges.txt'; 
 
## grab the contents 
my $content=get($url) || die $!; 
 
## grab the MD5 sum from our datafile 
my $oldmd5=''; 
 
if (-f $file) { 
  open(IN,"<$file") || die $!; 
  $oldmd5=<IN>; 
  close IN; 
} 
 
## generate the new MD5 
my $newmd5=MD5->hexhash($content); 
 
## same? 
if ($newmd5 ne $oldmd5) { 
  ## file has change ... do something 
  ## .... (insert your mailing code here)
  ## then save the new md5 
  open(OUT,">$file") || die $!; 
  print OUT $newmd5; 
  close OUT; 
}
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But to answer your question, change

if ($modified = $string) { print "the file is the same";
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to

if ($modified eq $string) { print "the file is the same";
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Update: The size of the files you are grabbing IS very important. 18MB is quite bit for casual downloading for comparison. In that case, I would also rely on the headers returned from HEAD (including Content-Length).