Darth Vader's true identity as Anakin Skywalker was a highly guarded secret, known only to a select few individuals including Palpatine, Obi-Wan Kenobi, Luke, Han, Leia, R2-D2, C-3PO, Master Yoda, Bail Organa, Ahsoka Tano, Darth Maul, Grand Moff Tarkin, Grand Admiral Thrawn, Jocasta Nu, Ferren Barr, Verla, Reva, ...
A duel between the Sith Lords Darth Vader and Darth Maul, the latter resurrected through Sith alchemy, took place on the moon Kalakar Six in the year 0 BBY.
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.
Maul even knew that Anakin had been groomed for a long time in order to become Sidious' apprentice. Maul's power in the Force gave him the ability to have intricately detailed visions of not only what was to come but also what had already occurred.
Did Yularen know Vader was Anakin? Of course he did, he was the head of the Imperial Security Bureau! He knew Anakin personally during the war, so he could recognize him easily. And it was his job to know who he was really dealing with so he could be alert for any signs of possible disloyalty from any angle.
Thrawn learned that Anakin Skywalker and Darth Vader were the same person during a mission with Vader, but kept the secret to himself. Thrawn's knowledge of Anakin's history explains his surprised reaction to hearing his name in Ahsoka episode 7 and likely gives him an advantage against Ahsoka Tano.
That includes the Siege of Mandalore, in which Anakin would have participated had it not been for the Chancellor's kidnapping. But their story began years before that, Darth Maul and Anakin's paths briefly crossed in Tatooine and later in Naboo.
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.
In the end, Vader reigns supreme against the reborn villain as his hatred drives him to be named the winner of the fight. In the end, Darth Vader's inner hatred and darkness made him the winner of the epic battle against Darth Maul.
As further highlighted by Vader and Kylo's redemption stories, it was a lesson that Kylo Ren never fully learned himself, which makes Darth Vader the more powerful user of the Force between the two of them in Star Wars.
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.
Ahsoka asks what he wants with Anakin Skywalker. Maul says that Skywalker is the key to everything. When Tano asks if it is to bring balance to the Force, Maul replies that Skywalker is the key to destroy and that he has long been groomed for his role as his master's new apprentice.
And while his warnings don't work, at the very least, it shows that Maul became aware of Anakin and what he would soon become. With the rise of the Empire, it's likely Maul caught on to Vader's existence pretty quickly.
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.
On the contrary, Tarkin was above Vader in the Imperial hierarchy. Vader had no real ranking, but since he was the Emperor's right hand man, he could often order officers around and, occasionally, even Moffs. Tarkin was something different.
Tarkin was without a doubt evil, no questions asked. But Tarkin never hunted, killed or tortured his enemies personally. He also never killed his subordinates for slightest mistakes. Vader did both things.
A Grand Moff received drastically increased authority, military forces, and funding as fitting their enhanced position. The position of Oversector governor was the sixth highest in the power structure after the Emperor, Supreme Commander, Grand Vizier, Advisors, and Grand Admirals.
