Sidious was well aware of how he killed his own master, Plagueis: in his sleep. Sidious needed to sleep, too, and during that time he was relatively vulnerable. Sidious intended to live forever so he could not give Vader (or anyone else) any opportunity to kill him. Hence, the Royal Guard acted as an insurance policy to help make sure Sidious could not be killed even when he was relatively vulnerable.
Also, the Royal Guards carried a large variety of weapons which were unseen underneath their capes:
Two members of the Royal Guard, covered head-to-toe in the blood-red armor indicative of their order, flanked the door. Each held a stun pole at station. Vader knew that each of their crimson capes hid a heavy blaster pistol, a vibroblade, and various other weaponry.
Lords of the Sith, p. 27
Some of these weapons have capabilities that Sidious did not have. In particular, a blaster pistol would probably have a longer range than even Sidious' Force Lightning.
Furthermore, the Royal Guards did prove themselves to be useful as bodyguards during the events of the novel Lords of the Sith, in which Sidious and Vader were attacked by Twi'lek rebels and a nest of lyleks on the planet Ryloth. When the attack began Sidious and Vader were theoretically quite safe on the Star Destroyer Perilous, but an attack forced them (and several Royal Guards) to evacuate the destroyer and crash-land on Ryloth. The danger was significant enough that the captain of the Royal Guard was killed defending the Sith Lords in combat.
It's also worth noting that Sidious' power as Emperor was justified in public in large part by the Jedi attack on him at the end of the Clone Wars. Sidious, known publicly as Palpatine, maintained a public persona as a kindly old man who was brutally attacked by the Jedi and was forced to (reluctantly) transform the Republic into the Empire and wield supreme power as the Emperor. Due to this attack, it was necessary for Palpatine to appear in public only with bodyguards (lest some rogue Jedi attack the poor, defenseless old man again!).
Not only were the Royal Guard necessary to maintain the fiction that Sidious was a defenseless old man who legitimately feared a Jedi assassination attempt, but the Royal Guard were among the very few who could be trusted to keep Sidious' secret:
[Vader] seldom saw his Master so publicly demonstrate his power. And he understood what it meant, of course. There must be no survivors who could bear witness. Only the Royal Guards could be allowed to live — only they could be trusted never to reveal what they’d seen, or even to talk about it among themselves.
Lords of the Sith, p. 168
and obey Sidious without question:
"Kill him," [Sidious] said. The leader of the Royal Guard, conditioned to obey any order of the Emperor instantly without question, did not hesitate. He stood, drew his heavy blaster, and shot his comrade once in the head, leaving a dark, smoking hole in his helmet.
Lords of the Sith, p. 148
As trusted lackeys the Royal Guards could be used to help keep Sidious' secret...and since the Royal Guard also obeyed Sidious without question, they could be ordered to execute anyone (even a fellow Imperial) who was not trusted enough to know Sidious' secret.
It's true that Sidious usually has little need of the Royal Guard when in private (especially on the Death Star), but why not use them in private as well as in public if he's gone through the trouble of conditioning them to obey him without question and reveal his secret to no one?