You can sue your neighbor, but you need to decide whether it's worth the time, money and effort (not to mention the additional stress of re-living an awful event over and over again).
If you win a lawsuit, the court will award you the value of the dog -- which is based on its market value, not what the dog is worth to you.
As a dog lover, I understand that it's difficult to put a price tag on your best friend. I just lost a 12 year old mutt from the pound who was priceless to me... but he wouldn't have been worth much in a lawsuit.
If the neighbor's dog is vicious, your time would probably be better spent filing a report with animal control so they can take appropriate measures to protect other pets and humans.
First you would have to prove that their dog killed your dog - was a report filed at the time with the police or animal control? Second, my colleague is correct - if you did sue and win you would only be awarded the replacement value of the dog and reimbursement for any vet bills. If you want to sue on principle, that's fine, but just be aware that you are not going to win much.