Your code works for me, once I fix it to actually become a program:

use strict;
use HTTP::Cookies;
use LWP::UserAgent;

# try to simulate wget:
my $cookie_jar = HTTP::Cookies->new(
                file => "mycookie.cookie",
                autosave => 1,
            );
my $ua = LWP::UserAgent->new;
$ua->timeout(2);
$ua->cookie_jar($cookie_jar);
$ua->env_proxy;

if( !@ARGV) {
warn "Requesting page";
my $request= HTTP::Request->new (GET=> "http://www.google.com/webhp?hl
+=en");
my $status=$ua->request ($request);
if ($status->is_success)
{
    print $status->decoded_content();
}
else
{
    print STDERR $status->status_line, "\n";
    return 1;
}

};

use Data::Dumper;
warn "Cookies in jar:";
$cookie_jar->scan(sub { warn Dumper \@_ });
[download]

It (re)generates the cookie file if run without any command line argument, and if run with a command line argument, it just dumps the cookies from the file.

Whatever your problem seems to be, I would suspect that it lies elsewhere.

Ok, I tested your script with google and it works.

Then it maybe related with tomcat auth issue:
I changed your script to work with my tomcat manager:

use strict;
use HTTP::Cookies;
use LWP::UserAgent;

# try to simulate wget:
my $cookie_jar = HTTP::Cookies->new(
                file => "mycookie-tomcat.cookie",
                autosave => 1,
            );
my $ua = LWP::UserAgent->new;
$ua->credentials ("host:8080","Tomcat Manager Application","user","pas
+sword");
$ua->timeout(2);
$ua->cookie_jar($cookie_jar);
#$ua->env_proxy;

if( !@ARGV) {
warn "Requesting page";
my $request= HTTP::Request->new (GET=> "http://host:8080/manager/statu
+s?XML=true");
my $status=$ua->request ($request);
if ($status->is_success)
{
    print $status->decoded_content();
}
else
{
    print STDERR $status->status_line, "\n";
    return 1;
}

};

use Data::Dumper;
warn "Cookies in jar:";
$cookie_jar->scan(sub { warn Dumper \@_ });
[download]

And then it printout cookie when I launch without parameters:

...
$VAR1 = [
          0,
          'JSESSIONID',
          '1F627DA1EED0B70CC94F4D3EB9ECD378.host',
          '/manager',
          'host.local',
          undef,
          1,
          undef,
          undef,
          1,
          {}
        ];

...
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but file of cookie is empty:

$ cat mycookie-tomcat.cookie 
#LWP-Cookies-1.0
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I discover in tcpdump / wireshark, tomcat is returning first an 401 error with "You are not authorized to view this page. If you have not changed ...", setting a cookie wich is used by lwp after (it try again the access and the page is loaded correctly), but seems like not saved in the file

Thank you !
PD: What is wrong with my question, so I'm getting negative reputation ?

Your cookie looks like this:

$VAR1 = [
          0,
          'JSESSIONID',
          '1F627DA1EED0B70CC94F4D3EB9ECD378.host',
          '/manager',
          'host.local',
          undef,
          1,
          undef,
          undef,
          1,
          {}
        ];
[download]

The HTTP::Cookies documentation says about the format:

  0  version
  1  key
  2  val
  3  path
  4  domain
  5  port
  6  path_spec
  7  secure
  8  expires
  9  discard
 10  hash
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The ninth parameter, discard is 1 in your cookie. Why do you think that HTTP::UserAgent should save this cookie into a file?

As to your "negative" reputation, your nodes are currently at "-1" and "0" respectively. I downvoted both of them, because you did not do show any relevant code first, and then did no work on making the problem easily reproducible outside your specific environment.