Why didn't Voldemort kill Snape with Avada Kedavra?

Question

Voldemort was very fluent in Avada Kedavra. He never ever hesitated before using that. Also, there was no thumbs down in using it. Avada Kedavra was quick and effective.

Why didn't Voldemort kill Snape with Avada Kedavra?

Answer 1

Because he thought Snape was the true master of the Elder Wand, and didn't want this to backfire.

"Why doesn't it work for me, Severus?" In the silence Harry imagined he could hear the snake hissing slightly as it coiled and uncoiled--or was it Voldemort's sibilant sigh lingering on the air? "My--my lord?" said Snape blankly. "I do not understand. You--you have performed extraordinary magic with that wand." "No," said Voldemort. "I have performed my usual magic. I am extraordinary, but this wand...no. It has not revealed the wonders it has promised. I feel no difference between this wand and the one I procured from Ollivander all those years ago."

Voldemort's tone was musing, calm, but Harry's scar had begun to throb and pulse: Pain was building in his forehead, and he could feel that controlled sense of fury building inside Voldemort.

Voldemort believes that because Snape killed Dumbledore, the Elder Wand's allegiance lies with Snape. Although Voldemort is currently using the wand, he notes that it doesn't behave as it should for him. By killing Snape, he hopes that it will change its allegiance to him.

However, he doesn't want this to backfire (as we see it does when he tries to kill Harry, the Wand's true owner, with it). It's reasonable to assume that the most powerful wand in history won't want to kill its owner. So, to be on the safe side, he uses another method, the one closest to him and easiest at the time - Nagini.

"The Elder Wand cannot serve me properly, Severus, because I am not its true master. The Elder Wand belongs to the wizard who killed its last owner. You killed Albus Dumbledore. While you live, Severus, the Elder Wand cannot truly be mine."

Other possible factors:

• He's hella mad and wants blood. In Voldemort's mind, Snape is in the way of what he needs. Being so close to killing Potter, only to discover that your servant has very inconsiderately claimed the wand you want, might make you angry. This fury may have made him more inclined to kill Snape, previously considered loyal, in a relatively gruesome way.
• On the other hand, maybe he's quite attached to Snape, and doesn't want to actually kill him himself. By making Nagini do it, he's offsetting the responsibility. However, this seems unlikely given his temperament - the passage says "He turned away; there was no sadness in him, no remorse."
• As pointed out by chirlu in the comments, JKR needed a way for Snape to reveal his motives, which he would only do close to death.

Edit: If this is the case, why would he try to use Avada Kedavra on Harry, when he knows that Harry is the true master of the wand?

I would argue that he does not really believe that Harry is the master of the Elder Wand. To be honest, I'm not sure that Harry believes it himself. Here's the passage in question:

“The true master of the Elder Wand was Draco Malfoy.”

Blank shock showed in Voldemort’s face for a moment, but then it was gone. “But what does it matter?” he said softly. “Even if you are right, Potter, it makes no difference to you and me [...] and after I have killed you, I can attend to Draco Malfoy...”

“But you’re too late,” said Harry. “You’ve missed your chance. I got there first. I overpowered Draco weeks ago. I took his wand from him.” Harry twitched the hawthorn wand, and he felt the eyes of everyone in the Hall upon it.

“So it all comes down to this, doesn’t it?” whispered Harry. “Does the wand in your hand know its last master was Disarmed? Because if it does . . . I am the true master of the Elder Wand.”
A red-glow burst suddenly across the enchanted sky above them as an edge of dazzling sun appeared over the sill of the nearest window. The light hit both of their faces at the same time, so that Voldemort’s was suddenly a flaming blur. Harry heard the high voice shriek as he too yelled his best hope to the heavens, pointing Draco’s wand:

“Avada Kedavra!”

“Expelliarmus!”

At this point, Voldemort is in the middle of a battle. He's not thinking as straight as he would be when planning to murder Snape - he came to Hogwarts, thinking Harry was dead, finds out he isn't, thinks he's still superior. His supreme arrogance still remains. Plus, Harry's argument is based on an exceedingly complex bit of wandlore - I doubt there would be a known precedent to base it on, either way. Note that he phrases it as a question - almost a test, daring Voldemort to call his bluff. Voldemort, at this point, is arrogant, angry, filled with adrenaline and confident enough in his stupidly complex plan. Nagini at this point is dead - he can't have her murder him. He also doesn't truly realise the protection Harry has granted himself and others by being willing to die for them, and therefore that Harry effectively has a backup shield. So, he takes the chance.

Answer 2

I have read Harry Potter and The Methods of Rationality which tries to be as close to canon as possible I think.

To be able to cast the killing curse you must hate that person enough that you would kill him with a knife when he is defenseless or be able to completely disregard the value of his life.

He might just not have hated him enough and put some value on his life.

As mentioned in this question about the requirements to cast Avada Kedavra

“Never used an Unforgivable Curse before, have you, boy?” she yelled. She had abandoned her baby voice now. “You need to mean them, Potter! You need to really want to cause pain — to enjoy it — righteous anger won’t hurt me for long — I’ll show you how it is done, shall I? I’ll give you a lesson —”

_{Harry Potter and the Order of Phoenix, ch. 36, The Only One He Ever Feared}