The module doesn't appear to provide a way to get the address with quoting and escaping removed so that addresses can be compared.

The following snippet does just that:

use Email::Address;

my @addresses = map { Email::Address->parse($_) } <DATA>;

print is_equivalent(@addresses) ? '' : 'not ' , "equivalent\n";

sub is_equivalent {
    my ($a, $b) = @_;

    return lc $a->address eq lc $b->address;
}

__DATA__
"John Doe" <jdoe@bla.com> (Johnnie "Two Toes")
jDOE@BLA.COM
[download]

It also doesn't disallow the very common user mistake of "everybody@gmail" (which can be valid as an e-mail address in some situations but isn't a valid address to give to somebody outside of your organization and so is worthwhile for us to disallow).

That's right. Any validation rules you want to impose beyond what RFC 2822 does is up to you, but Email::Address will tokenise the addresses in order to enable you to. Here's a snippet of how it deals with various address formats:

use Email::Address;

my @addresses = map { Email::Address->parse( $_ ) } <DATA>;

for my $address (@addresses) {
    printf("%8s: %s\n", $_, ( $address->$_ or '' ) ) for ( qw( origina
+l address user host name phrase comment format ) );
    print "-------\n";
}

__DATA__
abc@foo.com
bla@gmail
"Eve Rybody" <everybody@example.com>
foo@asdf.com%bar.com
"Alan B. Combs" <abc@foo.com> (I can't think of anything complex)
[download]

From there you can easily validate $username, $host etc.