I see Vadar's weakness is more psychological. He is in constant turmoil over the loss of his love, potential kids, and for what....too take orders from a guy who is always ready to replace him. He is not right in the head. Where Palpatine and Luke are 100% motivated by their own causes and completely in control. He is constantly in pain and thinking about the losses and the "What Ifs"

I think this is overall approach that was starting to be taken before the canon reset.

Not sure where it was, but I recall reading that Palpatine felt this was ultimately the case with Vader, that any limitations on his power were purely psychological in nature, with his extreme self-loathing over choices made and their outcomes being a major factor. Like you said, Palps and Luke (at least by RotJ) really don't have this extreme degree of internal conflict and are able to purely focus on who they are and what they need to do. When Vader gets pissed off enough to really focus his anger on someone or something, he's a force to reckon with because he's no longer being held back by his doubts and self-loathing. That's probably why he's so scary in Rebels because he's focused; having those walkers dropped on him didn't so much hurt him and get Vader mad, and an angry Sith Lord is a terrifying sight indeed. We never saw that kind of raw power in his confrontations with Luke because Vader was indeed struggling with his internal conflict, and thus not able to properly focus his power the way he'd probably be able to do against any other Jedi opponent.

Vader had two observable weaknesses in my opinon. He lacked ambision thus in the grand scheme of things though he weilded incredable influence; he actually owned nothing since would never be able to overcome his master. The second is, due to all the cybernetics he was especially weak against force lightning. But in all honesty he did have a disability, one so great that it made him sangificantly less then human.

His lack of heart; being more machine then man meant more then just the fact that he was only kept alive by technology but he was also a man who long stopped feeling. He did exactly as he was told and did little more then that, his few attempts at overthrowing in the EU always turned out to be a scheme of palpatines in the end. It meant that he would only ever be a master of evil, but not actually anything beyond that. He was a close minded brute who only thought about enforcing the emperors will.

That was the whole difference between him and Luke; Luke always had empathy and always had reasons for acting as he did, Vader was a heartless crude machine that could only murder. Theres a reason why Luke was able to best him quite handily on the Second Deathstar; he was able to grow and Vader was stagment. Sure he could crush the throat of a person with the powers of his mind, yet he couldn't resolve himself to become more powerful in himself, powerful enough to dislodge his own master as he instead killed commanders as he would ants. Very impressive, but hardly a challanging task for the lord of the sith. Palpatine put his power to more productive uses.

I firmly believe his presentation in the movies was fairly standard for that era; crippled adnormalties were seen as a sign of evil and Vader was essentially the embodiment of evil itself, so heavily modded up that he could no longer be even considered human or relateable. Like every Bond villian had a distingishable disablity of some description. Though that not always the case; in episode 6 he finally discovered his humanity, cast down palpatine and was finally able to look on his son with his own eyes. Of course the perception has largely changed since then, though star wars still continues to demonise their characters through cybernetics; Maul through excessive tattooing, General Grevious with the full body modifications and the villian in Episode 7 will likely continue the trend.

I think Vader is very emotional as presented in canon. You obviously only see that fixed mask but both Ezra and Ahsoka in Rebels sense his presence through the Force and speak of "the fear, the anger, the hate..." in tones of real astonishment at the strength of it. In Lords of the Sith (canon novel) there's a scene near the start where Vader is meditating and it's all about focusing his anger. Sith are really, really big on using their emotions. It's one of the chief dividers between them and the Jedi.

Even in the OT, I don't think Vader is unemotional. He just has a clipped, steady way of speaking. He sounds a lot like someone keeping a tight grip on themselves. And Leia remarks to Tarkin "I might have known I'd find you here, holding Vader's leash." That's not simply calling him his commanding officer, Tarkin is there to restrain Vader as Vader is always on the edge. Like when he chokes that officer who disparages his "sorcerous ways" and Tarkin has to stop Vader from killing the man right there in the boardroom. Vader is practically boiling with emotion. With a mask on, the actors' chief way of conveying the character's state of mind is through their voice and I think we all recognize Vader's voice as special. It's one of the all time great character voices alongside Orson Wells in Citizen Kane, Mark Hamill as the Joker and whoever played Jessica Rabbit as... Jessica Rabbit. And Vader's voice doesn't sound like someone emotionally dampened to me just because he's not shouting. He sounds like someone with dark oceans of cruelty and vengeance pinned behind a thin plaster wall of control.

Vader has a LOT of emotion.