In reality Bobby Darin was with Robert F. Kennedy during the campaign when Kennedy was shot, and in fact in the same hotel where/when it happened. "Beyond the Sea" had him in his trailer at Big Sur when he gets the news on the radio.
(at around 1h 27 mins) When Bobby Darin is in the hospital the first time, the heart monitor is a modern type with a flat panel addressable display. Monitors of this era used monochrome oscilloscope type CRT displays.
(at around 1h 7 mins) Sandra Dee tries to console Bobby Darin after losing the Oscar by telling him that Melvyn Douglas' win was "a sympathy vote" as he is dying, and his wife is a Congresswoman. Douglas died in 1981, and his wife Helen Gahagan served in Congress from 1945-1951.
(at around 1h 10 mins) When Bobby Darin can't start his car after the fight with Sandra Dee the car has only one headlight on. When he takes a golf club right after that and starts smashing the car, both sets of headlights are on, including the brights.
When Bobby Darin is singing his first song at the Copa, his hair changes throughout the song. First it's all combed back then there is a strand that is on his forehead, then it keeps flipping back and forth.
(at around 2 mins) In the opening scene of the movie in which Bobby Darin sings 'Mack the Knife' he tosses in an ad-lib by throwing a karate kick and shouting 'ah-so, Madame Myook.' This is based on an actual ad-lib which can be heard on a Nov. 10 1963 recording of 'Mack the Knife' at the Flamingo Hotel in Las Vegas. However, the real Bobby Darin says 'Madame NHU' and not 'Madame Myook.' Madame Nhu was considered the first lady of South Vietnam from 1955-1963. And, she was well known to the American public in the early 1960s because of her notorious political comments, staunch conservatism, and high sense of fashion.
Sandra Dee did not stay with Bobby Darin in the hospital as he was dying. She was in an alcohol induced denial at home. She was passed out on the floor from drinking and the family had to break into the house to find her and notify her. In fact, Bobby's second wife was banished from his room because she could not hold her tears.
The goof item below may give away important plot points.
Bobby Darin never, ever acknowledged Nina as his mother in public, and barely tolerated that knowledge in private. He also never believed Charlie was his father, nor did he appreciate anything Charlie did for him. This was a total fantasy "make-up" that Kevin Spacey did because he felt Bobby would have "done it eventually".