In season one, the Sparks have two children, Billy and Bobby. In later seasons, after the Sparks divorce, there is no mention of Bobby.
In The Guitarist Amplification (2009), Sheldon recounts big fights between his parents. Apparently, his mother once shouted that Jesus would forgive her if she put ground glass in his dad's meat loaf, and his dad was once on the roof skeet-shooting her Franklin-Mint collectible plates. Sheldon generally describes his father as a violent alcoholic several times on The Big Bang Theory. That does not gel with the portrayal of Sheldon's parents in Young Sheldon as a loving couple with occasional mild squabbles.
The Cooper family's church (First Baptist Church of Medford) looks different multiple times throughout the series.
Sheldon mentioned a few times in "The Big Bang Theory" that Billy Sparks was a bully of his in his youth. Billy Sparks (outside of intentionally scaring him with a chicken in the first episode for his own mild amusement) seems rather cordial with Sheldon in this series when they're together. However, Sheldon is often unreliable when it comes to his past. A great example is how he and Tam drifted apart. Sheldon saw it one way, and Tam took it a different way.
Dale Ballard has a poster of The Ballpark in Arlington (Texas) in his office but as season five, the timeline is only early 1992 and the Ballpark in Arlington did not open until April of 1994.
At the end of the credits, the Cooper family is shown standing in an arid landscape, which is typical of western Texas. However, their town of Medford is in eastern Texas, which is actually lush green and lacking in mountains.
Early in the series, Brenda's husband states that she's deathly afraid of chickens and doesn't ever go in the chicken coop. After he leaves her she says he took the chickens with him. But later on the chickens are still there and she goes into the coop to take care of them. She also brings Mary into the coop to hang out and talk in a later episode.
In the first season, Sheldon says that Mr Givens becomes the principal. However, Sheldon spends two years in high school and Mr. Givens is still a teacher.