Is this what you are looking for?
When the following is run...

use URI;
$u1 = URI->new('http://www.comp.leeds.ac.uk/Perl/');
$u2 = URI->new('http://www.comp.leeds.ac.uk/Perl/filehandling.html');
print "u1.host=",$u1->host(),"\n";
print "u2.host=",$u2->host(),"\n";
if ($u1->host() eq $u2->host()) {
   print "URL's host are the same\n";
}
else {
   print "oops\n";
}
[download]

It produces the follwing output...

u1.host=www.comp.leeds.ac.uk
u2.host=www.comp.leeds.ac.uk
URL's host are the same
[download]

It doesn't use URI::URL, but I'm not sure that you need to.

Update:
The URI perldocs have a section on parsing URI's with a regex. Here's a cut of that section...

PARSING URIs WITH REGEXP
As an alternative to this module, the following (official) regular exp
+ression can be used to decode a URI:

  my($scheme, $authority, $path, $query, $fragment) =
  $uri =~ m|(?:([^:/?#]+):)?(?://([^/?#]*))?([^?#]*)(?:\?([^#]*))?(?:#
+(.*))?|;
The URI::Split module provide the function uri_split() as a readable a
+lternative.
[download]

Update:
To get around the problems of case you could use the URI canonical method like...

use URI;
$u1 = URI->new('http://www.comp.Leeds.ac.uk/Perl/')->canonical();
$u2 = URI->new('http://www.comp.leeds.ac.uk/Perl/filehandling.html')->
+canonical();
print "u1.host=",$u1->host(),"\n";
print "u2.host=",$u2->host(),"\n";
if ($u1->host() eq $u2->host()) {
   print "URL's host are the same\n";
}
else {
   print "oops\n";
}
[download]