When Hercules destroys the marketplace near the beginning of the movie, the pillars start falling outwards from the center of the circle towards the entrance of the marketplace. In the next part of the scene, the pillars are collapsing inwards towards the center of the circle and fall on each other before they crash on the pottery vendor.
When Hercules throws the giant discus, the ground is rock; in the next shot, the ground is grass.
When Meg talks about her weak ankles, she sticks her left leg out in front of Hercules' face. In the next shot, it's her right leg that's stuck out.
In Meg's song "I won't say" during the phrase, "It's too cliché...", her eyes are blue instead of purple.
Five girls run in to mob Hercules, but only four run out.
All of the characters are referred to by their Greek mythological names except for Hercules. Hercules is the Roman equivalent to the Greek Herakles.
When Pain and Panic are disguised as little boys trapped under the rock, they say to call IXII. Hercules is set in Greece, and those are Roman numerals.
All numbers in the sequence "Zero to Hero" (in Hercules' credit card, when the muses give him a 10, "X") are Roman numerals. The Ancient Greek versions of the numbers would be alpha, beta, omega, etc.
As the planets align, Saturn is shown as the 4th planet. Jupiter, the largest planet, isn't even shown. Mars, being red, isn't depicted. The shape and sizes of the planets don't make sense.
To stop the eruption of a volcano, Hercules is shown plugging up the volcano with a giant boulder to stop the eruption. While this would temporally stop the eruption, it would get much worse and the pressure would build up inside until the volcano finally explodes causing a much bigger deadly disaster than ever and he wouldn't be able to save the city, from the inevitable disaster. Hercules would've been better off evacuating the people from the city.
Since this film is only an adaptation of the myths it is understandable for there to be quite a bit of artistic license taken with the stories. Any differences between this film and the myths belong in the trivia section of this site.
Hades primarily has flaming blue hair, which turns red when he gets mad. In the real world, blue fire is much, much hotter than red or orange fire. This may be because of blue generally being considered a calming colour, whereas red or orange are more associated with anger.
When Hercules is fighting the Hydra, he reaches into the ground and pulls out a slab of rock to throw at it. However, there's clearly no hole in the ground from where the slab came from as he does.
Mistake Hades' hair turns yellow, orange, and red when he's angry. Visible blue fire is the hottest flame, but also the cleanest i.e. no soot. The color changing could represent emotional soot blowing Hades composure.
Before becoming mortal, the infant Hercules's skin is much darker than his blond hair, which only turns red once he becomes mortal. Yet when he regains his godhood in the end, his appearance, hair and skin color are the same as they were when he was mortal.
When Phil admonishes Hercules, "Don't let your guard down because of a pair of big goo-goo eyes," he's clearly mouthing the word "blue" instead of "goo-goo." Meg's eyes were originally going to be blue, but later on they were changed to purple to match her dress.
After Zeus tells a joke, he falls into his chair saying, "I kill myself," but his lips don't match what he's saying.
Phil says he wants someone he trained to be in the stars. He claimed he trained Perseus, yet Perseus has a constellation.
When baby Hercules goes missing, Zeus assembles all of the gods and sends them on a desperate search to find Hercules. The story says that they found him, but could do nothing since he was mortal. Although Hades is in the underworld and busy working, one would think he would listen for any news about Zeus/Hercules. In addition to the plea and the news of the search outcome, wouldn't the fact that Hercules was alive be common-knowledge amongst the gods? If Hades had heard either the outcome or spoken to another god in the 18 years Hercules was mortal, he shouldn't have been surprised to hear that Hercules was alive. Also, being God of the Underworld, he could've checked for Hercules' soul to make sure that he was dead.