Although I don't believe it's ever explicitly stated, I think it's clear that the purpose is to give Sauron's name an air of mystique among his servants, thus instilling a greater fear of Sauron himself.

Consider the language that the Orcs in the service of Sauron use to describe the people above them in the hierarchy:

'Whose blame's that?' said the soldier. 'Not mine. That comes from Higher Up.

_{The Return of the King}

Ou:

'I'm not going down those stairs again,' growled Snaga, 'be you captain or no. Nar! Keep your hands off your knife, or I'll put an arrow in your guts. You won't be a captain long when They hear about all these goings-on. I've fought for the Tower against those stinking Morgul-rats, but a nice mess you two precious captains have made of things, fighting over the swag.'

_{The Return of the King}

"Higher Up," "They," "The Tower." By forcing his servants to use this vague language, Sauron is positioned as being so far above them that to challenge him would be unthinkable. If they were using his name constantly, they'd think of him as a person who could be challenged, not as a fixed, unchangeable System.

Note that this is common in fiction: for instance, in Harry Potter, Lord Voldemort intentionally sought to have his chosen name be feared, and certainly didn't permit his followers to use it. Arguably, this is the actual reason one is not permitted to say the name of God in various traditions, too: it would make God seem less divine and more mundane. This is another advantage of forbidding the use of his name: Sauron positions himself as a god. Only his expressly permitted emissary, or metaphorically his priest, can use his name. Indeed, if we consider Tolkien's Catholicism, Eru is perhaps literally the Christian God, and as such Eru's name is supposed to be treated with respect. By adopting this strategy for himself, Sauron mocks the power of Eru and the Valar.

In any case, Sauron wanted his name to be feared by the people of Middle Earth. I think he would have found a wicked-sounding name pleasing for this reason; indeed, it's very much in keeping with Tolkien's view of evil and corruption that someone would start off doing evil for good reasons, and end up embracing their evil reputation.

I think we can discard the alternative notion that Sauron objected to his name, and forbade its use for that reason. If that had been the case, he certainly wouldn't have ordered or even permitted his official spokesperson, the Mouth of Sauron, to use his name when addressing his enemies, and morever call him "Sauron the Great."

'These are the terms,' said the Messenger, and smiled as he eyed them one by one. 'The rabble of Gondor and its deluded allies shall withdraw at once beyond the Anduin, first taking oaths never again to assail Sauron the Great in arms, open or secret. All lands east of Anduin shall be Sauron's for ever, solely.

_{The Return of the King}

Surely the Mouth could have referred to Sauron obliquely, as "My Lord" (or "The Lord of the Rings") or "The Master of the Tower" or something along those lines, if Sauron had so desired. Moreover, had Sauron hated his originally derogatory name, he would surely have encouraged his servants to use one of his previous preferred names, such as Mairon, his name as a Maia, or Annatar, the Lord of Gifts. There's no evidence that he did so.

As a side note, out-of-universe, Tolkien's conception of his universe was always evolving. It's possible that at the point in time he wrote O Senhor dos Anéis, Sauron was never supposed to have had any other name. In the Silmarillion, for instance, there's no mention of Mairon; Sauron is used throughout, and Annatar is only mentioned as a pseudonym he used to disguise himself.

I believe @Adamant's answer is part of the reason, but it leaves out a more important factor.

Sauron was pretending to be dead.

No end of the Second Age, Sauron was in fact na maioria das vezes morto, with his body broken and his spirit dispersed. The hosts of Men believed he was todos dead, and Sauron was in no position to dispute the matter.

Levou ao longo dos anos 1000 before Sauron could exercise his will at all, another 1400 before he manifested at Dol Guldur, and still centuries more spent lurking there under the guise of "O Necromante". ("Nothing to see here, just an old fortune teller casting bones and talking to ghosts...")

If Sauron had announced himself during that time, the kingdoms of Men might have reunited against him under the Oath of Elendil. Men and/or Elves could destroy what little he had built, possibly driving him back to the shadows for millennia. Using pseudonyms wasn't just an air of grandeur, it was a necessary defense.

By the time he had rebuilt his forces in Mordor sufficiently to withstand his enemies, he had already been following this principle for many centuries. There was no reason to change it and plenty of reason to continue.

'Neither does he use his right name, nor permit it to be spelt or spoken,'

Perhaps it's too painful for him, after all that he has lost: His place among the Ainur; his stature as a second to Melkor, the integrity and beauty (such as it was) of his physical form - all gone. To now be referred to as "The Admirable" (Mairon) - must cut like a knife, even if that's said in reverence.

It is highly unlikely that anyone of Gimli, Legolas and Aragorn knows that Sauron's true/original name as one of the Ainur was actually Mairon. They believe that his true name is Sauron.

While in truth Sauron was a moniker given to him by his enemies in the First Age, meaning "The abhorred"/"The hated". It would make perfect sense to forbid the use of that name for this reason - he did not wish to be called the name given to him by his enemies.

Sauron kept referring to himself as Mairon or Tar-mairon throughout the First and Second age, and also took the name Annatar when living among Gwaith-i-Mírdain and forging the rings of power.

However, during the Fall of Numenor, Sauron was weakened and lost the ability to wear a fair face again (and perhaps the ability to at all shapeshift). Perhaps it would be painful for him to keep using the name he had during his previous charismatic incarnations, instead preferring not to be referred to directly at all by his minions. Who in turn didn't know what else to call him except Sauron or perhaps "the Dark Lord" or similar. He used O Necromante as an alias while he was still recovering his powers.

For sources see http://tolkiengateway.net/wiki/Sauron.

There's the old concept of the nome verdadeiro: If you know the true name of a person or a thing, that knowledge gives you power over them or it. For example, the Egyption goddess Isis manages to find out Ra's true name. That enables her to secure Ra's throne for her son Horus. In the Hebrew bible, next to all mentions of God have been redacted centuries ago with "pseudonyms" like "adonai" (meaning "Lord"). Because it would not only be blasphemous to speak God's (true) name, it would also give power over basically all of creation. And most of you have probably heard of the little imp in the fairy tale who could be defeated if you guessed his name correctly.

Many authors incorporated this idea into their works. In Ursula K. LeGuin's "Earthsea", a person's true name is revealed to them when they reach puberty. Telling somebody your true name is a tremendous sign of confidence, because that knowledge gives the other person magical power over you. In Christopher Paolini's "Eragon", everybody and every coisa has a true name in the "old language". Knowing those names also entails enourmous power. And when the last of of the "Nine Billion Names of God" is discovered in the eponymous short story by Arthur C. Clarke, the universe ends.

In Tolkien's works, Bilbo Baggins is very careful not to let Smaug learn his name. Because even he knows that it would be a very bad idea.

I don't know if Tolkien ever spelled it out with regard to Sauron, but we can probably assume that Sauron didn't want anybody to gain power over him by knowing his true name. So he didn't permit his underlings to use it.