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.
So, with Maul not really caring about Anakin, he thought it was a safe bet to escape. Unfortunately, from here onward, no confrontation between the two has ever been seen in any Star Wars media. However, that doesn't mean it never happened.
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.
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.
Yep. Darth Maul didn't really say or show straight up that he "feared" Darth Vader, but he openly stated his belief that he could not defeat Darth Vader without the help of the Jedi, therefore on his own.
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, ...
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.
The fight was orchestrated by a darksider cult called the Prophets of the Dark Side, who wanted Vader killed and replaced with Maul as Darth Sidious' only true apprentice, because they felt the former was "tainted" by his Jedi past and not worthy of being a Sith.
Darth Maul could have killed Anakin Skywalker at least twice and therefore prevented Darth Vader from being created, but that does not mean he would have been powerful enough to stop Darth Sidious' rise to power.
Having been trained to become Sith Lords, both Maul and Dooku sought to gain more power under Sidious while simultaneously looking to overthrow him. Toward the end of the Clone Wars, Dooku and Maul did actually meet each other in canon in the Son of Dathomir comics by Jeremy Barlow.
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.
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.
R2-D2 was present when Padmé gave birth to Leia and Luke, so he knew Luke was Anakin's son – but what he didn't know was that Anakin became Darth Vader. In addition to that, Anakin's fall was surely a painful event in R2-D2's life, one that he definitely wouldn't like to remember or even talk about.
Later, Maul orchestrated the siege of Mandalore as a means to draw Anakin Skywalker and Obi-Wan Kenobi to him with plans to kill Skywalker and deprive Darth Sidious of his new apprentice.
After a thousand years in the shadows, Maul was the first Sith to reveal himself and kill a Jedi Master. So, assuming the fight was over, he may have dropped his guard. Because of this, he couldn't sense that Obi-Wan was planning to jump behind him, and it ended up costing him the fight.
He just wasn't explicitly taught it, or dark side sorcerery more generally. His training was focused on the skills of a Sith assassin, being an overwhelming lightsaber duelist. You need real legs to be able to force-siphon the electric current from the ground. That's why Maul or Vader can't use lightning.
Like his master, Darth Sidious, better known as Sheev Palpatine, Maul's main weakness was his overconfidence. In Luke Skywalker's confrontation with Palpatine during The Return of the Jedi, he tells the Emperor that his overconfidence is his weakness.
Darth Maul and Darth Vader never met in the Star Wars canon, although Maul and Anakin Skywalker were heavily involved in key events of the Clone Wars. That includes the Siege of Mandalore, in which Anakin would have participated had it not been for the Chancellor's kidnapping.
