Just before the launch of the ship, Cookie complains about having to carry non-essentials on the trip. One of the non-essentials he mentions is "cilantro", however, in 1914 this seasoning would have been referred to as "coriander leaves".
In several scenes, the submarine's crew are seen wearing gas masks akin to those developed for trench warfare. Gas had not been deployed as a weapon yet, but the concept of a gas mask dates back to the 9th century. A gas mask was patented in America in 1849 by Lewis Haslett, for filtering dust. Furthermore, the African American inventor Garrett A. Morgan invented a gas mask in 1912 that proved useful for gas/smoke-filled environments. Military use of gas masks did not occur until at least 1915, but Rourke runs a private army with unconventional equipment, to say the least. It is not difficult to imagine him or an associate seeing the need for protective masks in a dusty, potentially gas-filled underground environment.
In the scene where Milo is handed the Book of The Shepherd you see Mr. Whitmore's huge aquarium, containing some Coelacanths. In 1914 the Coelacanth was thought to have been extinct since the Cretaceous period, first live specimen was caught in 1938 in Chalumna River.
When Milo and Kida are trying to figure out how to power to the vehicle using Kida's crystal, the crystal is removed from around her neck to power the vehicle. In the next shot the crystal is around her neck again and still powering the vehicle. And again in the next shot the crystal is missing, and replaced again in the next shot.
In Whitmore's room, when he's explaining why he's funding the Atlantis trip, he smacks the painting above the mantelpiece with his cane at his head is level with, or above, the mantelpiece. However, in the next shot, his head is at least a foot beneath the mantelpiece.
When the group is settling at the campsite, when Milo asks Audrey about her life, in one scene she's taking her boots off. Note that she has only an overall and a sleeveless shirt. In the next scene, one can see her arm throwing her sock away, and she's wearing a sleeved shirt.
In Helga Sinclair's initial introduction scene ("Milo James Thatch...") she is wearing a bracelet, and is lacking her trademark gloves. In the next scene, the bracelet is gone and she is wearing said gloves.
In the beginning where Milo is leaving the basement to go up for his meeting, he picks up all his maps, and has them under one arm, but then when he picks up the picture of his grandfather, his hands are completely free, arms empty.
When the travelers first arrives at Atlantis, they all step in front of the large "digger" machine to get a better look at the city. Then we see a wide shot of the city and travelers and the digger is nowhere to be found.
While on the highway to Atlantis, several times Milo has to take off at a run to escape something and he never has anything but his clothes with him. Yet when he reaches Atlantis he still has his book and his satchel though he didn't have them before.
When Milo is rehearsing his proposal in the museum basement, he rubs against the chalkboard and wipes the map he drew onto his clothes. He then stands in front of the chalkboard and the chalk map on his clothes matches the missing part of the map, when in fact it should be a mirror image.
The expedition's submarine, the Ulysses, is apparently powered by steam. This would be impossible for any vessel traveling underwater. The fires from the boilers would consume all of the air in the vessel within a fairly short period of time, leaving the vessel without power and the occupants asphyxiated.
Cookie tells Audrey that he has 'all 38 United States' as a tattoo. In 1914 the United States consisted of 48 states, but Cookie probably got his tattoo years before, maybe when the United States did consist of 38 states - between 1877 and 1890.