I think what you want is their RSS Feed (technically you are not supposed to re-use that either, but don't tell anybody I told you). Just use XML::Simple to parse it for what you are looking for and show your own rain images.

Welcome at the Monastery!
There is a better way to do this: instead of grabbing that page, build your own page using data got via RSS! Grab the data available from one of the sources listed at Yahoo RSS using XML::RSS::Feed and build the page you want with HTML::Template. Have fun!

Looks like there are only 48 images there, and the following code should grab all of them for you.

use LWP::UserAgent;
use strict;
use warnings;

my $ua = LWP::UserAgent->new();
for my $index (0 .. 47, 3200) {
    print "downloading $index.gif\n";
    my $res = $ua->get("http://us.i1.yimg.com/us.yimg.com/i/us/we/52/$
+index.gif");
    if ($res->is_success) {
        open GIF, ">", "weather$index.gif";
        binmode(GIF);
        print GIF $res->content;
        close(GIF);
    } 
}
[download]

Update: Added 3200.gif, as ikegami pointed out.

(above caveats on harvesting the images) but just for sake of example here's a bash one liner:

  for n in `seq 0 99` ; do wget http://us.i1.yimg.com/us.yimg.com/i/us
+/we/52/$n.gif ; done
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Also, instead of get() and open/print/close, can just use LWP::Simple's getstore() method:

use LWP::Simple;
use strict;
use warnings;

for my $index (0 .. 99) {# actually 47 is the last index
    print "downloading $index.gif\n";
    my $res = getstore("http://us.i1.yimg.com/us.yimg.com/i/us/we/52/$
+index.gif", "weather$index.gif");
}
[download]

or just:

perl -MLWP::Simple -e 'getstore("http://us.i1.yimg.com/us.yimg.com/i/u
+s/we/52/$_.gif", "weather$_.gif") for 0 .. 99'
[download]

Unfortunately your Perl code does not work. I called binmode() in my code for a reason. Your code would be fine with html files, but with images, it does not work.

If you test your code, you will see that some images are broken (or cannot be recognized as the right format).

You missed 3200.gif. Their meanings are listed here.