If he had used the Avada Kedavara curse, he'd have to use the Elder Wand. He believed if he had used the Avada Kedavara curse, the wand would have resisted him doing so as it did when he tried to kill Harry who is the true owner of the Elder Wand. To avoid this, he slit Snape's throat and then let Nagini attack him.
Apparently, he thought that the true allegiance of that wand was with Snape.
Voldemort: Where does its true loyalty lie?
Snape: With you... Of course, My Lord.
Voldemort: The Elder Wand cannot serve me properly, because I am not its true master. The Elder Wand belongs to the wizard who killed its last owner. You killed Dumbledore, Severus. While you live, the Elder Wand cannot truly be mine.
It's safe to assume that the most powerful wand in the world wouldn't kill its owner. Therefore, one has to kill or defeat the current owner in another way to become its true owner. We already saw that the owner of Elder Wand was murdered in the story of three brothers told by Hermione and when Draco disarms Dumbledore by Expelliarmus, the allegiance of this wand changes.
Hermione: Harry! Why didn't it work for him? The Elder Wand?
Harry: It answers to somebody else. When he killed Snape, he thought the wand would become his. But the thing is... The wand never belonged to Snape. It was Draco who disarmed Dumbledore that night in the Astronomy tower. From that moment on, the wand answered to him. Until, the other night when I disarmed Draco at Malfoy Manor.
Hermione: So, that means...
Harry: It's mine!