Factual error: In the middle of the movie, when Alexander is traveling through time and the landscape around him is evolving, the geologic sequence makes no sense. It shows sand and dust covering what was once New York, then water forming into a river and wearing down the gorge, and then a glacier covering the area. The glacier would have exerted hundreds of millions of tons of pressure on the land, smoothing out the gorge that had worn down through the sedimentary rock. The glacier would then have melted into the hollow it had formed, much like the Great Lakes. It would have made more sense for the glacier to have come first, then retreated and melted to form a river that carved out the gorge.
Factual error: When Alexander makes his first trip into the past, you can see the second hand of the pocket watch turning backward. The hand is jumping second by second. This is a mistake. A quartz watch (battery powered watch of our times) would do it this way, but a mechanical one from 1903 would turn backwards the same way it turns forward: 5 steps per second, which for the untrained eye looks like a smooth movement.
Factual error: While we can only speculate about the imagined physics that allows the time machine to operate, we see that when Alexander accidentally puts his hand outside the protection of his machine's 'force field' during travel to the future, the fingernails grow by at least five millimetres before he snatches his hand back in pain. This amount of growth indicates that his hand has experienced at least a month while outside the field. Since his blood vessels inside the field would only be supplying a few seconds worth of blood, his hand would be starved of oxygen and therefore dead and well into decomposition by the time he manages to get it back within the field. His only option would be immediate amputation to prevent gangrene in his arm.
Factual error: The motor-carriage that is seen outside the skating rink and later on the street by the flower shop, has current bicycle type tires (metal rims and spokes). Most vehicles of the time had wooden rims and spokes, lined with rubber.