A Sith decides to use his real name or Sith name depending on which would be more effective and useful for the Sith Order. Different individuals have different skills, so some Sith Lords are more effective as overt Sith whereas others are more effective as hidden Sith.
Palpatine and Dooku
Palpatine/Sidious was a skilled politician who instigated various crises in order to gain power.
The first such crisis was the Trade Federation's invasion of Naboo. As Darth Sidious, Palpatine ordered the Trade Federation attack Naboo. Meanwhile, Palpatine was Naboo's Senator in the Republic Senate. Palpatine used the Republic's failure to protect Naboo from the Trade Federation to unseat the Supreme Chancellor at the time (Finis Valorum) and acquire the office himself. The fact that Palpatine's home planet was invaded no doubt helped him gain sympathy votes for his election as Supreme Chancellor.
Once he became Supreme Chancellor, he instigated the Clone Wars. The ongoing emergencies of the Clone Wars allowed him to acquire new powers for himself as the Republic's Supreme Chancellor. Finally, at the close of the Clone Wars, the Jedi discovered he was secretly a Sith Lord and attempted to arrest him. Palpatine's face was disfigured when the Jedi attempted to arrest him, but he was able to label the Jedi as traitors and order them executed. Palpatine was able to proclaim himself Emperor while portraying himself as a kindly old man brutally attacked and disfigured by the power-hungry and traitorous Jedi.
Palpatine was far more effective as a politician with a hidden Sith identity than a Sith combatant. He was able to gain and keep power primarily through political crises which garnered sympathy for him. He would not have gained such sympathy nor any political office in the Republic had he publicly used his Sith name and overtly presented himself as a Sith Lord.
Dooku's primary purpose as a Sith Lord was to serve as the leader of the Confederacy of Independent Systems (CIS) in order to instigate the Clone Wars. As previously mentioned, Palpatine used the Clone Wars to gain more power for himself. Dooku was an effective leader for the CIS since he was charismatic, wealthy (he was the Count of Serenno), and politically well-connected (he was a former Jedi Master). As a political leader, it was important for Count Dooku to emphasize his noble heritage from Serenno rather than openly call himself the Sith Lord Darth Tyranus.
Maul and Vader
On the other hand, Maul and Vader were both more effective as terrifying, overt Sith Lords.
Maul's tattoos make him look evil, making it impossible for him to pull off the "kindly old man" persona used by Palpatine or the dignified, noble persona used by Dooku. Maul therefore embraced his persona as a fearsome Sith warrior. Maul concentrated on learning martial arts and mastered the double-bladed lightsaber, making him particularly well suited for assassinations and lightsaber duels.
Vader was similarly terrifying due to his mask and armor, his use of Sith powers like Force Choke, and his similarities with General Grievous:
It was that genuflecting obedience, the steadfast devotion to execute whatever task the Emperor assigned, that had given rise to so many rumors about Vader: that he was a counterpart to the Confederacy’s General Grievous the Emperor had been holding in reserve; that he was an augmented human or near-human who had been trained or had trained himself in the ancient dark arts of the Sith; that he was nothing more than a monster fashioned in some clandestine laboratory. Many believed that the Emperor’s willingness to grant so much authority to such a being heralded the shape of things to come, for it was beyond dispute that Vader was the Empire’s first terror weapon.
- Tarkin, p. 73
In Vader's case in particular, his (former) Jedi identity as Anakin Skywalker conflicted with his Sith persona. Anakin was a hero for the Republic in the Clone Wars, but Vader was the Empire's enforcer and terror weapon. Vader therefore had to hide his identity as Anakin.
Furthermore, both Vader and Sidious wanted Vader to forget his former identity as Anakin Skywalker. Vader wanted to forget his painful loss of Padme and therefore needed to forget his former life as Anakin. And Sidious knew that Vader needed to forget his life as Anakin so that Vader would not consider Luke to be his son and would remain loyal to the dark side. Indeed, Luke's response to Vader's claim that "Anakin" no longer had any meaning for him is:
It is the name of your true self. You've only forgotten. I know there
is good in you. The Emperor hasn't driven it from you fully. That is
why you couldn't destroy me.