When where Bernstein was conducting the choir, at 58:30, when he turned his head, you can see the cigarette in his mouth. Before, and after that moment, he was holding it in his hand.
It is never explained how if Bernstein is the one telling the story, he knows certain events happened (like his wife speaking with their daughter) that either a) he wasn't present for or b) the movie doesn't say how he was informed of the event.
The day after Bernstein makes his wildly successful debut with the N.Y. Philharmonic in November of 1943, the story is carried on the front page of the N.Y. Times. One of his friends notes that the front page also includes a headline reading "Hitler Bombs Poland." Germany had bombed and conquered Poland in September, 1939, so the country had already been under German occupation for over four years at the time of Bernstein's debut concert.
In spite of the meticulous attention to detail everywhere else in this film, the fact that Bernstein's eyes were brown and not blue was overlooked.
Near the end, Bernstein listens to REM's "It's the End of the World as We Know It" in his car, a song that mentions him by name (as well as various other famous figures with the initials LB). It was released in 1987, three years before his death, so it is entirely possible that Bernstein heard it.
When Leonard Bernstein and his two buddies pull up to a country house with a dog in a convertible, you can see a Taste of the Wild dog food bag being carried by the younger man. That brand of dog food didn't come out until around 2007.
When Felicia, Shirley Bernstein and Jamie Bernstein have tea in the Palm Court of the Plaza Hotel in New York, in a scene set in the early 1970s, the room's huge stained-glass dome is clearly visible multiple times. The dome was not added until the hotel's 2008 renovation, however.
When the Snoopy parade float passes their window, both the float and their apartment window appear to be on the same level of the trees. It appears to be a first-floor apartment, but parade floats are not at street level.
Even in old age, numerous pictures attest that Leonard Bernstein's nose wasn't nearly as big and bulbous as the prosthetic Bradley Cooper is wearing.