In the movie, Brian Wilson is portrayed as obese, bearded, and refusing to leave his bed as early as 1967, listening to Sgt. Pepper's Lonely Hearts Club Band on headphones all day. In reality, despite his Smile project being scrapped and even though he wasn't as involved in the Beach Boys as he had been before, Brian was still very much a part of the group, very actively writing, producing, and performing on the records until the mid-1970s.
The movie depicts Carl Wilson, Brian Wilson, Bruce Johnston, and Mike Love singing "God Only Knows" live in the studio with the musicians. In reality, Brian Wilson (and most music producers) never records "live" - he has the instrumental track recorded and perfected first, then later overdubs the vocals; this was definitely the case with "God Only Knows." Also, Mike Love does not appear on "God Only Knows;" the background vocal that Mike is shown laying down in the movie was in reality provided by Brian Wilson.
The film implies that under Murry Wilson's direction, the Beach Boys recorded "Barbara Ann" in a standard session in 1963 or 1964. In reality, the song was recorded "unplugged" in September 1965 after Dean Torrence (of Jan and Dean fame) visited the session for the Beach Boys' Party! album and suggested that they record it. By that time, Murry Wilson had long since been fired as the Beach Boys' manager.
In the film, Dennis Wilson is shown fully-clothed and disoriented as he walks from the street to the dock before stumbling into the water at the marina and drowning. In reality, the day he drowned, Wilson was clad only in cut-off jean shorts and had been diving in search of a photo he'd previously thrown overboard from his yacht.