An oldie but goodie.
Alright, so let's look at the scene carefully:
Voldemort raised the Elder Wand, holding it as delicately and precisely as a conductor's baton.
'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.''
'All this long night, when I am on the brink of victory, I have sat here,' said Voldemort, his voice barely louder than a whisper, 'wondering, wondering, why the Elder Wand refuses to be what it ought to be, refuses to perform as legend says it must perform for its rightful owner .. and I think I have the answer.
Harry Potter and the Deathly Hallows - pp.525-7 - Bloomsbury - Chapter 32, The Elder Wand
So, this isn't a question of Voldemort being unable to use the wand, in the way that Harry is unable to use the blackthorn wand. He is able to perform his extraordinary magic just as well as he ever could but he doesn't sense anything special about it.
Why is this, in his opinion? Why does the wand refuse to be what it ought to be, why isn't it revealing wonders? Because he hasn't earned the wand's respect. He believes true mastery of the wand comes through murder and bloodshed. Then it will not just work it will make him invincible.
Now then, why would Voldemort think that the blithering old fool Albus Dumbledore had won the wand's respect? After all, legend has it that the Elder Wand is "a wand more powerful than any in existence: a wand that must always win duels for its owner" (Harry Potter and the Deathly Hallows, p.331). Dumbledore was an immensely powerful wizard, and yet when he duelled with Voldemort in Harry Potter and the Order of the Phoenix, Voldemort came off the better.
Why, then, would Voldemort think that "Dumbledore was the last person to wield the elder wand with its full allegiance" merely because he wielded it at all. He had taken possession of it, but need he have won its respect in the Dark Lord's mind? Need he have taken possession of its full powers and range of abilities, merely by winning it from Grindelwald non-lethally? I would say no.
But Voldemort wasn't looking for merely another wand. His old, cherished wand had not defeated Potter. Just another wand, with a different core to Potter's wand hadn't worked. He needed something stronger, something special. He needed the Elder wand as a way to guarantee victory over "the one with the power to vanquish the Dark Lord", the one who will "have power the Dark Lord knows not". He needed everything it had to give.
'He wanted you to tell him how to overcome the connection between our wands,' said Harry.
'You told him about the twin cores? You said he just had to borrow another wizard's wand?'
Ollivander looked horrified, transfixed, by the amount that Harry knew. He nodded slowly.
'But it didn't work,' Harry went on. 'Mine still beat the borrowed wand. [...]'
'We were talking about the other wand, the wand that changes hands by murder. When You-Know-Who realised my wand had done something strange, he came back and asked about that other wand, didn't he?'
'Yes, he asked,' whispered Ollivander. 'He wanted to know everything I could tell him about the wand variously known as the Deathstick, the Wand of Destiny or the Elder Wand.'
'The Dark Lord no longer seeks the Elder Wand only for your destruction, Mr Potter. He is determined to possess it, because he believes it will make him truly invulnerable.'
Harry Potter and the Deathly Hallows - pp.400-1 - Bloomsbury - Chapter 24, The Wandmaker
He needed its respect. He needed to kill the last person to merely possess it, the last person to merely make it work, he needed to show that he was worthy of the full abilities of the Elder Wand.
And so Snape had to die.
P.S.: Needless to say, I'm suggesting what Voldemort may have thought, given what he knew and believed. I'm not suggesting that Snape really did have to die according to the actual lore of the Elder Wand. I'm trying to show how Voldemort could have believed that Snape had to die, even though Dumbledore was able to use the Elder Wand without having murdered Grindelwald.