When Jules and Verne fly the train through the big Courthouse billboard (the one that says "Historic Hill Valley"), the train flies directly through the clock. The hole is the same size as the clock. Yet in the next shot the hole has taken up half the Courthouse!
When Jules and Verne give the newspaper to the security guards, the front page with the story about the disaster is now in the middle of the paper.
Doc and Marty travel to South America in the 1500's to look for a toad that has since become extinct. They refer to the toad as Bufo marinus. In fact, this is the Latin name for the cane toad, which is far from extinct.
After they managed to get Franklin to do his kite expiriment and fix the past back up, wouldn't the chairs that Jules and Clara burned come back into existence?
When Franklin did his kite experiment, his son William was with him. But William Franklin is nowhere to be seen in this episode.
We see the Courthouse twice in this episode, and both times the clock is still working, not frozen at 10:04 PM like it should be!
Verne's name is spelt "Vern" on the scoreboard near the start of the episode.
Marty reacts to being called "chicken" by Bifficus and as such enters the race. Marty learnt his lesson at the end of Back to the Future Part III and should not react to being called "chicken" anymore.
Doc couldn't repair the train because he needed a flux capacitor - but he was somehow able to send the little truck forward to 1992 with the note for Marty.
Marty crashes into the barn door, creating a hole. But when he leaves for 1692 a few minutes later, the hole is gone.
When the train leaves to come Back to the Future just before the flux capacitor blows up, it is in the sky, but when it lands in Salem, Massachusetts, it is on a set of train tracks.
Doc brings his children to the cretacious period (which ended 65 million years ago), however the date he imputs into the time machine is 3000000 B.C. some 62 million years off!
When Lord Biffingham and his assistant fall down the stairs of the tower, they leave Clara's dungeon door open. Surely she could have made an attempt to escape whilst Biffingham was busy falling down the stairs?
In 1991, McDonald's also released four Happy Meal toys, featuring Marty on his hoverboard, Doc in the DeLorean, Verne in his Junkmobile and Einstein in the time train
Verne Brown's (one of Doc Brown's kids) middle name is revealed in this episode. It is Newton, after Sir Isaac Newton.
Verne says Jules was using an Etch-A-Sketch when he was 2. Jules was probably born in 1886 or 1887, and I'm pretty sure that by 1888-1889, they didn't have an Etch-A-Sketch!
Ed. note: But they're time travellin' kids! That doesn't really count. FYI, the Etch-A-Sketch was first produced by the Ohio Art Company on July 12, 1960. (link)
Response: True, they are time travelling kids, but back in 1888/9 when Jules was 2, Doc had not finished building the time train. He probably took about 10 years to build it. So, without a time machine, how could Jules get the Etch-A-Sketch?
Ed.: Once a time-traveller, always a time traveller. There's an unwritten time travel story in there somewhere, not a goof. But, I'll leave up this debate; visitors to our pages may decide for themselves...
When Verne heads back to 1864, the time circuits say "February 11th 1864". Verne arrives in the evening of the 11th. When Doc, Marty, Jules and Einstein find out where Verne went, they also input the Destination Time in the train as February 11th 1864 - yet arrive the day after Verne got there.
When Doc uses the time train to enter the year 1864, the train is flying when it travels trough time, but when it arrives to 1864, the train is on tracks, not flying.