Qui-Gon JinnThough Yoda and Qui-Gon respected each other, they were often at odds due to their differing beliefs; Yoda looked to the future while Qui-Gon lived in the moment.
Arguably, Qui-Gon is wiser in some ways, given that he knows how to preserve his consciousness in death. Yoda is wise, but sometimes flawed, given that some of his views align with the problematic thinking of the Jedi of this era.
Thus, while Qui-Gon was less powerful than the likes of Yoda and Mace Windu, Palpatine feared him more. In the simplest terms, he was a wild card, and with the Sidious executing an intricate decades-long plan to seize power and destroy the Jedi, a wild card was something he just couldn't afford.
In Revenge of the Sith it is made known that "dying AND maintaining one's identity" is not in fact what happens to every Jedi, but an ability acquired by Qui-Gon's spirit and passed on to Yoda and Obi Wan.
Yoda Talks To Qui Gon Through The Force!(Deleted Scene)
Why did Qui-Gon tell Yoda to go to Dagobah?
Qui-Gon explained that he had been tasked with guiding Yoda along this new path, teaching him to retain his consciousness after death and commune with the living. After Order 66 and the declaration of the First Galactic Empire, Yoda returned to Dagobah, living simply amid its many creatures.
Moreover, throughout the duel, Yoda continually adapts to and even matches Palpatine - forcing his enemy to constantly attempt new tactics. Palpatine clearly hated Master Yoda with a fierce passion, but he was under no illusions that the Grand Master would be easy to defeat.
Master Qui-Gon Jinn's words echo in Vader's mind. There's no anger, nor hatred in this memory – Vader simply recalls one of the first lessons the Jedi Master slyly gave to young Anakin Skywalker.
Whether Vader thought of Qui-Gon specifically was not mentioned. However, it was safe to say he most likely did. Vader felt regret over how he disappointed everyone who cared about him, like Obi-Wan, Padmé, Yoda, etc.
The late Master's legacy suggests that he may have actually been a Sith working against the Jedi Order all along, successfully orchestrating the downfall of the Jedi and the rise of the Empire.
Sidious believed himself to be far above any Jedi. Until this moment: It was then, that he realized they were relative in power. He was by no means afraid of Yoda.
Qui-Gon Jinn was the wisest of the human jedi. He was as knowledgeable in the force as any of his jedi predecessors and he was the first to commune with Master Yoda from the netherworld of the force. A power master Yoda then passed on to Obi-Wan.
As a wise and well-respected Jedi Master, Qui-Gon Jinn would surely have been named to the Jedi Council if only he hadn't routinely gone against its wishes.
Palpatine himself seemed to have a positive relationship with his mentor Darth Plagueis, and while he tortured Darth Vader to bring out his strength, he also gifted him Mustafar as a show of respect, having first offered his own home planet of Naboo to his apprentice.
Summary. Palpatine's true source of power comes from his intense hatred for everyone in the galaxy, which only fuels his strength. Not only that, but the more people hate Palpatine, the stronger he becomes as well, making inspiring hatred a crucial part of increasing his strength.
Dooku loved Qui-Gon like a son due to their many similarities; both were individualists, idealists, and both had knowledge of the Force that was unique.
We are led to assume this as in TCW Qui-gon admits his training was incomplete. He could only appear on Mortis, anywhere else he could only be a disembodied voice.
