Apollo 8's Translunar Injection (TLI) burn is shown with the spacecraft pointed at the moon. TLI is always performed parallel to the surface on the opposite side of the earth from the moon.
In this episode, Jim Lovell accidentally deploys the life vest in the Apollo 8 command module on the way to the moon. After some discussion of what to do, Frank Borman, a graduate of West Point, says in a gently teasing tone "Navy man." Bill Anders repeats the remark, also in a somewhat denigratory tone. This would most likely not have been something Bill Anders would have said, as he was a 1955 graduate of the Naval Academy, similar to Jim Lovell, who graduated in 1952. It is unlikely that a fellow USNA grad would disparage, even in teasing, a fellow USNA grad.
The Lunar Orbit Insertion burn is shown as starting before the capsule loses sight of Earth. In fact, LOI took place when Apollo 8 was directly behind the Moon.
The astronauts are shown reacting to Earthrise on their first orbit around the Moon. In fact, the command module was generally not positioned to see the Earth. It was only on the fourth orbit when commander Frank Borman was executing a roll maneuver that it happened to be pointing toward the Earth, and Bill Anders took the famous photograph.
The Trans-Earth Injection burn is shown as ending moments before Earth becomes visible. Like the LOI burn, it would have taken place when the spacecraft was directly behind the Moon.
The soundtrack includes several songs from the Allman Brothers Band. The band was not formed until 1969 and its first album was not released until November of that year.
A crew member can be seen briefly during Tom Hanks' opening monologue. What appears to be an arm is visible from behind the right side of the sculpture of Apollo. The crew member moves out of sight just as Hanks finishes speaking.
As the spacecraft rotates following the TLI, the sun comes into view and each of the three astronauts is shown looking directly into the light. This would not have happened as they knew that this would have blinded them.