Why does Aro want the Cullen family exterminated so badly in last twilight book

In the Twilight series, Aro, the leader of the Volturi, does not express a desire to exterminate the Cullen family. While there is tension between the Volturi and the Cullens due to the latter's unique vampire abilities and refusal to join their coven, Aro ultimately spares the Cullens in the final book, "Breaking Dawn," after being convinced of their peaceful intentions and witnessing the strength of their newfound allies.