When Captain Kirk tells Lieutenant Palamas to spurn Apollo, as she stands up and walks away from him, she has some kind of red mark on the side of her nose, like a scar or an indent from wearing glasses, yet in the next scene with Apollo it is gone completely.
Apollo and others constantly refer to the Trojan War time period as "5000 years ago." Since the Trojan War happened in the 13th century BCE, this would place Star Trek's setting sometime in the 37th century CE. All other references throughout the series indicate a date of no later than the 31st century, before the writers settled on the 23rd.
At minute 22, Spock refers to Apollo by name. Apollo told his name only to the landing party, and not the people left on the ship.
Scotty says that the hand is putting 800 gsm on the hull of the enterprise and was in danger of being crushed. He reports the pressure as 800 csm. Assuming CSM stands for Centimeters per Square Meter, that converts to only 0.0011 PSI. Unless the Enterprise's hull is made of tin foil, that pressure is nothing. The formula is 1 gram per square centimeter to pound/square inch [gauge] = 0.01422 pound/square inch [gauge].
Lt. Palamas says that in ancient literature, Apollo was the son of the god Zeus and a "mortal" named Leto. All known classical references state that Leto was a Titaness, a member of the elder gods.
When Apollo is mentioning names, he consistently uses Greek mythology. But then he says "Hercules". Hercules was the Roman name. The proper Greek would have been Heracles.