See pp. 174--185 of "Perfect Rigor", a biography of Perelman, by Masha Gessen, for the case that he has Asperger's. I think the case looks plausible. Quote from the book: "I spent an hour on the phone describing Perelman to Baron-Cohen, the famous psychologist volunteered to fly to St. Petersburg to evaluate the famous mathematician-- who sounded very much like many of his clients-- thus joining the long list of people who had volunteered help that Perelman did not welcome." (Page 178)
From page 179: "Virtually everything people have recounted to me about Perelman's behavior... fits the typical picture of a person with Asperger's syndrome. His apparent disregard for conventions of personal hygiene is common to Aspergerians, who perceive it as a nuisance forced upon them by the incomprehensible world of social mores. The trouble he had with articulating his solutions to problems is also classic. `People with Asperger's often put in far too much detail,' said Baron-Cohen. `They don't know what to leave out. They are not taking into account what the listener needs to know.' That is the theory of mind problem: the point of telling is not to get the point across, but solely to tell. Schoolmates tole me Grisha was always willing to answer questions about mathematics; the problem arose if the questioner did not understand the explanation. `He was very patient,' a former classmate recalled. `He would just repeat the exact explanation, again and again. It was as though he could not imagine that somebody found it hard to understand.' She was probably exactly right: he really could not imagine it."
A possible meltdown is noted on page 184. It accompanied his resigning from the Steklov Institute over getting paid too much!
A summary from the book of Baron-Cohen's results on the relation of mathematical skill and Asperger's: "When he (Baron-Cohen) tested his theory on a population of Cambridge University undergraduates, it turned out that the mathematicians among them were three to seven times more likely than other students to have a diagnosis of an autistic condition. Baron-Cohen also developed the AQ, or the autism-spectrum quotient, test, which he administered to adults with Asperger's or high-functioning autism as well as to randomly selected controls and Cambridge students and winners of the British Mathematical Olympiad. The correlation between math and autism and/or Asperger's was proved again: mathematicians scored higher than other scientists, who scored higher than students in the humanities, who scored roughly the same as random controls."