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.
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, ...
Maul rants at Vader, emphasizing that he is driven by seemingly unparalleled hatred. This seems to give him an edge over Vader, whom he disparages for being part-machine. Darth Vader cuts Maul's double-sided lightsaber in half, but Maul is able to disarm Vader.
Considering his infamy, it's hard to deny that Darth Vader is the more powerful of the two, but his victory over Maul wouldn't come easily. As the Emperor's enforcer, Vader is terrifying.
Star Wars Rebels lead creator Dave Filoni even admitted that Maul and Vader were meant to have a fight, but he felt it would take away from the battle with Ahsoka. But as things stand, Star Wars not only lacks a fight between Vader and Maul, but the two have never even spoken to each other.
Darth Vader FIGHTS Darth Maul who Calls him Anakin
Could Maul beat Vader?
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.
Obi-Wan faced off against Darth Vader four times and physically beat him twice while beating him through his Force development in a third fight. Obi-Wan Kenobi had the high ground that led to Vader's disfigurement and becoming the ultimate Star Wars cyborg.
It's already been shown in the source material… more than once… that Darth Maul is not on Darth Sidious's level. Every time he dueled Sidious as his apprentice (in Legends) he got easily defeated. In the show Star Wars: The Clone Wars, Palpatine takes on Darth Maul and Savage Opress simultaneously.
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.
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.
Maul always knew. He figured it out before everyone else. He even staged a massive war on Mandalore with the sole intention of luring Anakin and Obi-Wan there. Maul saw through his former Master's plan, realising that Anakin was the apprentice Sidious sought.
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.
Yoda is extremely wary of the dark side, he knows that it clouds judgment. Maul is a prime example of this. Yoda would easily be able to outmaneuver Darth Maul and then outsmart 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).
Infil'a put up a fierce fight against Vader, almost killing him by throwing the already damaged Sith Lord off of a cliff. Despite severe damage to his suit, Vader survived the fall, and repaired himself with parts from Infil'a's droid, Arex.
This isn't a fight, this is a curb-stomp. Kylo would absolutely never decline a fight with Darth Vader, a force of legendary power whose surpassing will certianly gain him a place among the most powerful of the Dark Side, but he is clearly outmatched in power.
The only Sith Lord who actually killed Vader, Darth Sidious kept his apprentice in line for over two decades. His true advantage was Force lightning, a power which would wreak havoc on the suit keeping him alive.
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.
He used Maul as bait to lure out his mother and kill her, and when Maul escaped Sidious no longer considered him a real threat. Maul's hatred and bitterness against Sidious (and the Sith as a whole) spurred him to find ancient Sith artifacts to use to exact his revenge.
