Character mistake: Almanzo says he hasn't seen his brother Royal in 10 years, but in fact he saw him not long ago as he looked after his 2 sons who were terrors, and at that time Royal didn't even have a daughter.
Factual error: When the hot air balloon comes down with Carrie in it you can see N410AA printed on it. This is commonly called a tail number on planes and is issued through the FAA, which of course did not exist during the time period of the episode.
Other mistake: At the very end, Mr. Hanson is seen walking toward the church. Because of his stroke, he's slow and unsteady. We see several close-ups of his feet, showing how he's struggling. Two of those shots are the same footage. It's obvious, because you see that he drags his foot over a small stick in the road and stumbles - and in both shots, it is exactly the same.
Continuity mistake: Charles is delayed getting home on his wedding anniversary because a wheel on his wagon broke, he had no jack and he had to make repairs during a heavy rainstorm. Rain is still falling late at night and can be seen through the window when Caroline hears Charles arriving. She runs outside to greet him and suddenly there is no rain.
Factual error: Laura Ingalls Wilder was born in 1867. Ralph Waldo Emerson died in 1882, which means she was at the maximum 15-16 years old (and already married) in order to meet him IF he were to appear in 1882 which is not possible because: The problems with his memory had become embarrassing to Emerson, and he ceased his public appearances by 1879 (according to wikipedia).
Continuity mistake: At the end, when Lars Hanson hears the church bell ringing, he gets up from his bed and goes to look out the window. When he first gets up, his hair is completely tousled - definite "bed head." The shot changes to show him leaning out the window, and his hair is neatly combed. When the shot returns to the inside of the room, his hair is once again rumpled and untidy.