Maul was originally loyal to Sidious, obeying his every command without question. Maul and Palpatine's bond as master and apprentice originally was strong enough that Maul knew about the plan to instigate the Clone Wars, as he expressed disappointment upon finding it had begun without him.
However, not everyone loved Maul. In fact, Darth Sidious didn't care much for him. Here's what the evil Sith Master really thought of Darth Maul, as outlined in Star Wars: Hidden Empire #1 (by Charles Soule, Steven Cummings, Victor Olazaba, Gur-eFx and VC's Travis Lanham).
Palpatine despised Darth Maul for not being "quite good enough" and lost interest in him once his gaze turned towards Anakin Skywalker. Palpatine saw Maul as a pawn and used him to his advantage, but never considered him a direct threat to his life.
Sidious revealed to Maul that he did not intend to kill his former apprentice, as he still had use for Maul; it was Sidious' hope that Maul would lead the Sith to Mother Talzin, Maul's mother and the leader of the Nightsisters, so she could be eliminated as a potential threat to the eventual Sith rule of the galaxy.
During the days of Star Wars: Revenge of the Sith, Maul had a deep hatred towards Palpatine and wanted to tear down his plans. So, it would seem logical to publicly expose his old master. But two things were standing in Maul's way: he was beyond terrified of Palpatine and knew the Jedi wouldn't trust him.
Doubtful. Even with his power, Palpatine couldn't anticipate every single thing that occurred. He had only just discovered Anakin and while he had plans to make him his knew apprentice, that was along time ahead. If he really wanted Darth Maul out of the picture then he would have done it himself.
He regretted it because he had essentially wasted 20 years training him. But since he was an enemy at the time of his death on Tatooine, Sid was glad that he did not have to keep watching his back anymore now that the loose ends were tied off.
Overall, Dooku was far more equipped to become a Sith Master than Maul was. Had he managed to overthrow Palpatine, Dooku would have been able to use his position, combat abilities, and dark side Force powers to take his place on the throne Palpatine had carved.
The two were able to free Maul after he had been interrogated by the Sith Lord Count Dooku, and together the three rendezvoused with more Death Watch forces on the moon of Zanbar. Maul took command of his forces once again, and assured them that the war they had longed for was at hand.
What happened to Darth Maul after Sidious beat him?
After defeating Maul and killing his brother Savage Opress, Sidious takes him to a Separatist prison, where Count Dooku tortures him for information about the Shadow Collective. Prime Minister Almec arranges Maul's escape and the latter then heads back to Zanbar to command the Death Watch army.
After finding out that the Queen had escaped the blockade, Darth Sidious was furious. He sent his Sith apprentice, Darth Maul, to find the Queen and the Jedi, destroy the latter, and bring the former back.
Up to Star Wars: Episode VI - Return of the Jedi, Palpatine treats Vader poorly, and they very much have an abusive relationship. Darth Vader hates Palpatine yet is eager to please him, whereas Palpatine goes back and forth between acting as a friend and showing his hatred.
Regardless, based on the season 7 of The Clone Wars, and the fact that Darth Maul was aware of Anakin Skywalker's future and his role as the Chosen One, it is very likely that Darth Maul indeed feared Darth Vader to some extent, and preferred to avoid a direct confrontation with him, at least, on his own.
Seeing his Master murdered caused Obi-Wan to press his attack aggressively, catching Darth Maul off guard. When Maul regained his composure he was able to Force-push Obi-Wan into a melting pit, where he held on for dear life. Calming himself, Obi-Wan leaped over Maul and used Qui-Gon's lightsaber to sever Maul in two.
Sensing his potential, Sidious took Maul as an apprentice while he was still a child in 40 BBY. Anointed by his Sith Master as Darth Maul, the Dathomirian was raised to hate the Jedi Knights that served the light side of the Force.
Following Qui-Gon's death by Maul's hand on Naboo, Dooku visited Palpatine in distress about his apprentice's death, hoping to have one day made Jinn an ally while also directly stating he knew of Maul's existence.
By the time of Episode 3; Anakin is in his prime, and already one of the Jedi Order's most powerful “masters” (although technically still a knight), just behind Obi-Wan, Windu, and Yoda. Maul wouldn't stand a chance. Anakin stomps.
Mace Windu vs Darth Maul | Fandom. Mace windu clear winner 9/10 times. His use of vapad would make the fight play out similair to windu vs palpatine. He would use the seething rage pouring out of maul against him, and as mauls force power was so much weaker than windu's it would be an undisputed win for windu.
