Even though they were both trained by Palpatine, Dooku and his predecessor, Maul have been enemies every since his mind was restored by Mother Talzin. This is because the two hated each other. Dooku despises Maul for being a threat to him and Maul hated Dooku for being his replacement.
dooku spoke highly of darth maul and refered him as a “great sith lord” in tcw episode monster.
Yes they did. Windu considered this a strong possibility during Son of Dathomir, upon hearing that Maul and Dooku had made contact and were fighting.
In one of the canon comics Sidious outright implied that Maul was pretty much head and shoulders above Dooku when adressing the potential. While Dooku was a useful pawn for him, Maul in the other hand seems to be actually his first choice as a heir before Anakin was really part of the picture at least.
Maul rivals Dooku in saber skills and is physically superior to the Count. Though DOoku is in better shape, is stronger, is faster, and more durable than most men his age (there are numerous sources of this just ask and I'll explain).
Maul was tracked to Zanbar by General Grievous, the leader of the Separatist military, and a Separatist fleet. Dooku ordered Grievous not to kill Maul, simply to defeat and wound his forces enough that Maul would seek out Mother Talzin's help.
After learning that Palpatine was secretly the Sith Lord Darth Sidious, Dooku turned to the dark side and became a Dark Lord of the Sith himself, replacing Darth Maul as Sidious's second apprentice. His new master bestowed upon him the Sith title of Darth Tyranus.
Another contender for the weakest Sith could be Count Dooku, or has he's also known Darth Tyranus. Dooku was a cunning and talented Sith Lord, known for his swift skills with a lightsaber and his ability to manipulate populations.
His former colleagues in the Jedi Order saw Dooku as misguided but a man of principles, and refused to believe he could be an advocate of violence and war. They were wrong; Dooku had sworn allegiance to the Sith Lord Darth Sidious, taking the name Darth Tyranus.
Sidious broke the Rule of Two at least twice: once by training Darth Maul while still under the tutelage of Darth Plagueis; and secondly, by taking on two apprentices at once, and in the form of two Jedi: Count Dooku, the former Padawan of the legendary Jedi Master Yoda, dubbed Darth Tyranus, and the Fosh Jedi Vergere.
Though he saw Anakin's potential, Yoda could not ignore the danger inherent in Anakin's unrestrained emotional attachments. To teach Anakin responsibility, Yoda assigned Skywalker a Padawan learner: Ahsoka Tano. Yoda knew there would come a time when Anakin had to let go of Ahsoka.
Bottom line, Dooku never considered himself a Sith. And he wasn't; he was a man looking to change the galaxy in a way the Jedi were unwilling to try. And that is why he joined the Sith and became Darth Sidious' apprentice, Darth Tyrannus. When Dooku became Sidious's apprentice, he was already an elderly man.
Mission to Mathas. Padawan Windu was trained in the ways of the Force by Jedi Master Cyslin Myr. Following his early trials as a youngling, Windu attained the status of Padawan by becoming Jedi Master Cyslin Myr's apprentice.
A Force-sensitive Human born on the planet Junction V in 102 BBY, Lorian Nod was trained in the ways of the Force by the Jedi Order at the Coruscant Jedi Temple. As an Initiate in the Temple's academy, Nod was sorted into a clan of his peers where he became quick friends with fellow student, Dooku.
If Maul knew, he would have tole Obi Wan or another Jedi during the Clone Wars. Dooku knew Palpatine was Sidious as he told Obi-Wan about him controlling the senate and referred to him as the dark lord. There's no real way of actually knowing if he knew ab palpatine.
For him, finding and killing Maul for his failure simply wasn't worth the effort. In the end, Palpatine was right, but his carelessness and disregard for attachment proved to be his undoing when Darth Vader's bond with his son Luke was stronger than the Emperor could ever possibly imagine.
The Grand Master's apprentice
At one time, Dooku was the Padawan of Yoda, the Grand Master of the Jedi Order. In 86 BBY, Dooku later completed his Initiate Trials and hoped to be selected as the Padawan apprentice to Lene Kostana due to her mission in hunting down relics associated with the dark side.
Studying Jedi Archive records, Dooku based his new weapon design after the curved hilts that were standard during the heyday of Form II lightsaber combat. The curve allowed the hilt to fit better into his hand, allowing for superior finesse and precise blade control.
1 Darth Sidious
There has been no Sith, as powerful, meticulous, nor as evil as the Emperor himself, Darth Sidious. As previously mentioned, Sidious trained in the Dark Side by Darth Plagueis before Sidious killed Plagueis in his sleep, making Sidious a Sith Master.
Revenge of the Sith (2005)
Anakin gives in to his hatred of Dooku and uses his anger to overpower him, severing both of Dooku's hands and leaving him helpless.
Truly, the most powerful Sith Lord was Darth Sidious, better known in his public persona of Chancellor (later Emperor) Palpatine. Through cunning and manipulation, Sidious killed his master to claim the mantle of Dark Lord of the Sith.
Anakin regrets this decision, and Palpatine reasons with him saying that Dooku was too dangerous.
Conclusion. Since that unfortunate day on Naboo, Maul hated Obi-Wan for sending him into a life of suffering and disappointment. Maul spent years plotting his revenge and was able to get a portion of it by killing Duchess Satine in front of Obi-Wan.
Later stripped of his black Nightbrother markings, Maul was left solely red-skinned. Agonizing Sith tattoos were applied personally by Darth Sidious in an attempt to cleanse Maul's mind as well as demonstrating Maul's complete dedication to the dark side.