In Star Wars canon, Grand Admiral Thrawn appears to be an infallible, near-perfect character. Although there are people in the Empire who hate him because of his race or because he was seemingly handed his position for no reason, most of the Imperial higher-ups laud his competence. Besides in Force-related situations, there seems to be no tactical event that Thrawn cannot handle. Having seen him in Star Wars Rebels and his canon trilogy of books, he seems to be the best at everything. He has been proven to be able to use a lightsaber. He has a vast array of languages at his disposal. His only total failures were Force-related, as they could not be addressed tactically. Is there some fatal flaw that Thrawn has the I missed? Does Legends Thrawn have difference flaws than canon Thrawn?
This answer is from a Legends perspective; I haven't seen Rebels and I don't know much about the Canon incarnation of Thrawn. Unmarked spoilers ahead for the Thrawn trilogy and the first three or four chapters of Specter of the Past.
In addition to Thrawn's own actions, I'm also going to talk about Grodin Tierce, one of the antagonists of the Hand of Thrawn duology; he was a student of Thrawn's, and his ability to pass his tactics off as Thrawn's is a major plot point. In addition (warning: major spoiler for Vision of the Future),
he was created by Thrawn to serve as a successor of sorts. Tierce is a clone, and some of Thrawn's memories and thought patterns were used to create the cloning template, so he not only uses the same tactics Thrawn does, he thinks about them the same way.
The first and most fundamental flaw Thrawn has is elitism. He probably wouldn't call this a flaw as such; Chiss society is (self-described as) highly meritocratic, so he assumes that he and other who earned high stations are meritorious, and those "stuck in" low stations aren't.
One consequence is that he never really gets a handle on Talon Karrde. He understands Karrde's cunning and skill, but he seems to miss his capacity as a leader and the loyalty he inspires in his followers. One of the few outright mistakes he makes in the original Thrawn trilogy is underestimating the loyalty Mara Jade feels towards Karrde, which leads her to defy Thrawn and sets both of them firmly on the New Republic side of the conflict.
Tierce has a similar (lack of) relationship with his co-conspirators, including Moff Disra, Disra's pirate ally Zothip, and their impersonation artist Flim. This doesn't just lead to more antagonism in their plotting than is strictly necessary, his dismissal of Zothip opens up a key avenue of investigation for the New Republic. (Also, Zothip tries to have Tierce shot, but that's not a major threat as such things go.)
Another consequence of this is that Thrawn and Tierce hold their secrets very tightly. Thrawn teaches Pellaeon and Tierce, but no one else. He dismisses the opinions of the Imperial fleet and intelligence teams but he never bothers to elucidate why except to his chosen students. Compare this to other notable strategists like Wedge Antilles, Admiral Ackbar, or even the later Czulkang Lah, who are quite willing, even eager to teach others to do what they do.
Second, and related, Thrawn has a habit of only planning for one scenario, the one he feels is most in keeping with the nature of the participants. He's a perfectionist, which isn't uncommon for Chiss. That serves him well most of the time - such as at Bilbringi - but if he misjudges a person, he can't effectively plan for their actions, and he lacks redundancy.
The most serious example of this failing is the mistake that would eventually kill him: his misjudgement on Honoghr. His analysts brought him information that the commando Khabarakh's ship had been tampered with by Wookiees following his mission to their homeworld. This presented Thrawn with two basic conclusions: that Khabarakh had failed and been captured, and the irregularities with his present situation were to cover that dishonor, or that he'd been compromised or turned, and the irregularities were to cover his defection. He believed the Noghri's loyalty was unshakable and handled the situation accordingly, without planning for the second possibility. As a result, he completely missed the entire Noghri plot against him.
Interestingly, Tierce also missed the Noghri plot. He recognized that the Bilbringi assault would be an ideal time for an assassination attempt, but he believed it would come from without rather than within because he thought Luke would be the key actor.
Third, Thrawn is a brilliant tactician but is progressively less adept at higher levels of strategy. He really only has the one grand strategy: find the biggest threat, divide them, crush the parts, and repeat. That's not a bad grand strategy, but it's not as imaginative as the tactics and operational-level strategy he comes up with in support of it. (It also got him kicked out of the Chiss, because he couldn't or wouldn't adhere to their isolationist grand strategy goals.) Tierce had exactly the same problem, which was bad because his Empire was in no way able to sustain that conquest the way Thrawn's could. Pellaeon tells Tierce off to his face about this: although he's tactically brilliant, he lacks the vision to win the war.
Even at the level of campaigns and theaters of war, Thrawn tended to be fairly staid and unimaginative. For instance, his divide-and-conquer tended to focus on the big names in the New Republic, such as the Mon Calamari and the Bothans. In contrast, both Warlord Zsinj and (much later) Nas Choka preferred to pursue less important species and landed several blows before the New Republic even realized it was being divided. A similar idea, using pirates to wear down the New Republic's flanks, was almost literally handed to both Pellaeon and Tierce, but both rejected it, partially because of the idea of using "lowly" pirates. (Pellaeon, of course, also wanted to make peace, seeing it as the only reasonable course of action.)
While this post is a few years old, the new 'Ascendancy' books have added some new flaws into canon. While we saw it a bit in the 'Thrawn' books, 'Acendancy' gives us some concrete examples of Thrawn's blindspot. He has a tendency to downplay or misinterpret the social and political consequences of his actions. In 'Chaos Rising' we get flashbacks of an event in his early career where he encounters a conflict between two alien races. He gets around the rules in classic Thrawn fashion by using the targeting lasers on his ship to secretly transmit the tactical weaknesses of the attackers to the defenders. It works, and he manages to protect an important trade port and save some lives.
What he doesn't account for is the political fallout. The aliens that he 'helped' take the information and go on the offensive, settling old scores and destablizing the region. He only barely manages to avoid having the ascendency sucked into a conflict they have no interest in joining.
We also see that he is oblivious to the political maneuverings within the Ascendency, just as he later would be in his imperial days. He is working for the greater good of his people and succeeding. He simply can't imagine how that would make him a threat and inspire people to work against him. The only reason he manages to maintain and advance his position is because he has allies who DO understand the political situation and shield him from the worst of the fallout.