Short version: psychological warfare.
Voldemort hates his muggle name, as Slytherincess points out in an answer to a related question:
'You think I was going to use my filthy Muggle father's name for ever? I, in whose veins runs the blood of Salazar Slytherin himself, through my mother's side? I, keep the name of a foul, common Muggle, who abandoned me even before I was born, just because he found out his wife was a witch? No, Harry. I fashioned myself a new name, a name I knew wizards everywhere would one day fear to speak, when I had become the greatest sorcerer in the world!'
Chamber of Secrets Chapter 17: "The Heir of Slytherin"
"Voldemort" is more than a name, it's an identity. It's a way of elevating himself above the people he hates, and past he hates more. "Voldemort" is powerful: the Dark Lord, the most feared wizard of all time, practically a god in the eyes of his followers, saviour of the Purebloods. "Tom Riddle" is powerless: a small, frightened, unloved, half-blood child.
Using that name effectively does two things:
- It reminds Voldemort of how utterly powerless he is; no matter how hard he tries, he's still that frightened child
- It rejects Voldemort's self-written mythology, and rejects his power to impose that mythology
Neither of those things are conducive to Voldemort's continued mental stability.
It's also probably because Dumbledore always did:
'It was foolish to come here tonight, Tom,' said Dumbledore calmly. 'The Aurors are on their way -'
'By which time I shall be gone, and you will be dead!' spat Voldemort.
'You do not seek to kill me, Dumbledore?' called Voldemort, his scarlet eyes narrowed over the top of the shield. 'Above such brutality, are you?'
'We both know that there are other ways of destroying a man, Tom,' Dumbledore said calmly
Order of the Phoenix Chapter 36: "The Only One He Ever Feared"
He does it again in Half-Blood Prince, even after Voldemort protests:
Dumbledore stood and swept over to the cabinet where he now kept the Pensieve, but which then was full of bottles. Having handed Voldemort a goblet of wine and poured one for himself, he returned to the seat behind his desk. "So, Tom... to what do I owe the pleasure?"
Voldemort did not answer at once, but merely sipped his wine. "They do not call me 'Tom' anymore," he said. "These days, I am known as —"
"I know what you are known as," said Dumbledore, smiling, pleasantly. "But to me, I'm afraid, you will always be Tom Riddle. It is one of the irritating things about old teachers. I am afraid that they never quite forget their charges' youthful beginnings."
Half-Blood Prince Chapter 20: "Lord Voldemort's Request"
This is just another type of psychological warfare. Dumbledore is often called "the only person Voldemort ever feared", so using Dumbledore's mannerism is another way of getting inside Voldemort's head.