Queen is used to buff in party, and has a low PAtk.
I would refer you to http://www.l2pets.com , however the site appears to be down at the moment. I hope the summoner community hasn't decide to close up shop.
On topic, at level 56 your Kai has about 900 patk and quenn has about a 325 patk so you can not solo or even do any real damage with Quenn. Good if all you want is to be a buffer for your archer parties. But I rather fight than hit a buff icon every 2 minuets.
basically Kai is a solo summon, high Pattk and good HPs.
Queen is used in parties, as she can give two fairly decent buffs and only takes up 5% of your exp. however her pattk is about a third of Kai's, and her HPs are about half.