The Olympic Coven is a gifted and unique family. Aro, being the greedy person he is, covets the gifted members, such as Edward and Alice. He's afraid that one day, the Olympic coven will be stronger than the Volturi, and more dominant. Especially with Bella and her shield coming into equation now, they are the ones rendered powerless.
They want Alice, Bella, and Edward. They think Renesmee will not be able to control her thurst and thus for expose the vampires. The Volturi is the most powerful group of Vampires because of the most members special talents because of Bella's power they are weak against the Olympic Coven. So destroying them would put them back on top and not have the fear of being over taken. The Volturi do not like how the Cullens don't abide by the laws and are trying to find ways around them. Such as, exposing vampires to Bella and then not changing her, having an alliance with werewolves, and having an immortal child.
the main reason is carlisle when he left it was because they have no respect for the human being and when carlisle was in the volturi they tried to make him a human blood sucking leech but he refused so now maybe they are seriously getting back at him or they are now really trying hard to convince him (something like that)
The Volturi don't fight the Cullens because Alice shows Aro a vision of the fight. The Cullens kill the Volturi and their witnesses. Leaving his reason of the fight, Renesmee, alive. Aro then is frightened of the vision and leaves, giving the Cullens victory.