Percy comes off as more knowledgeable in Greek mythology, whereas his book counterpart stood in as an Audience Surrogate and newcomer to Greek myths.
Sally constantly regaling Percy with Greek stories when he was younger likely contributed to him paying more attention to them — since his mom seemed to enjoy them.
Percy is even seen playing Mythomagic in Episode 1 where Grover bonds with him over the game. This could further explain Percy's knowledge.
In Episode 2, when Percy first meets Luke Castellan in the Hermes Cabin, there's a sort of ominous theme playing as Luke approaches Percy, who initially believes Luke is someone who wants to give him a hard time like other bullies he's encountered. The ominous soundtrack hints that Luke isn't who he seems to be and, ironically, Percy was right that Luke will eventually be giving him and his friends "a hard time".
It could be due to Annabeth not travelling outside of camp for so long, she's unsure about what candies to buy at the gas station, so she gets them all. But she could also be wanting to get on friendlier terms with the boys, hence buying more than enough food to feed all of them.
Annabeth looking particularly annoyed at Grover trying to sing the "consensus song" could be due to the fact he tried that before when guiding her, Thalia and Luke before.
When Echidna sets the Chimera on the three, she takes a moment to mock Annabeth by saying she wouldn't know what a good mother would do for her children. At first, this just seems like a simple villainous taunt about Annabeth not knowing her mother, but with The Reveal that Athena removed her protection from her temple and was potentially the one who even sent Echidna after them in retaliation for mailing them Medusa's head, she had personal knowledge that Athena is not really a good mother.
Poseidon having a Papa Wolf streak in both mythology and the book series isn't the only reason he'd intervene so dramatically to save his son from Echidna. After all, who was responsible for setting up this fight but Athena, his longtime rival? Saving his son and showing up his rival as the better parent at the same time is simply a perfect set of circumstances for the trident-wielding Olympian.
Grover names the Turbot War and Lobster War as two examples of wars he is a fan of. Both those "wars" were bloodless conflicts that were disputes over fishing rights. Of course Grover being a Satyr would know about those, since if limits on fishing are not put in place, it leads to over-harvesting of fishing grounds, ecological disasters, and extinction of native fauna, which was a very real concern in both conflicts and one of the reasons why the local governments were trying to get foreign vessels out of what they saw as their territorial waters.
Ares being annoyed at how close Athena is to her owl and how it symbolizes her wisdom isn't just sibling jealousy on his part, but it also is a perfectly reasonable thing to question. Contrary to popular belief, owls are pretty unintelligent birds that often make huge life-threatening blunders that other birds are smarter at avoiding, with only their exceptional hunting skills making them distinct from other members of the avian family. Thus, Athena lording over Ares while talking to her owl probably annoys the God of War because aside from hunting strategies, there's very little wisdom that it could offer to someone who's supposed to be the goddess of wisdom.
The "What is Love" song is not only used as Hephaestus questioning his marriage with Aphrodite, but also his relationship with his mother Hera. Ever since he was brought into the world, all he's known is rejection and betrayal by the people who should love him, but don't. His mother tossed him off a cliff due to his disfigurement, and his wife is having an open affair with Ares. So, all he can do is channel those feelings into his craft and make sure he's known as someone "not to be pushed around" to protect himself.
In the beginning of "We Take a Zebra to Vegas", the opening shows Percy dreaming of Kronos' meeting with the Lightning Thief in the form of Yancy Academy in the principal's office, with the "principal" talking with disappointment to a "student" about how said "student" almost failed him.
It would make sense because, in Episode 1, the principal's office was where Grover "betrayed" Percy by getting him kicked out of school, and how Percy felt powerless towards an authority figure. "You shall be betrayed by one who calls you friend."
The dream also manifests as Percy's greatest fear: being betrayed by someone he could trust and look up to while being looked down by an authority figure.
Grover meets another Satyr at the Lotus Casino who claims to be close to finding Pan, and he's more right than he thinks he is; Hermes seems to be a regular guest at the Lotus Casino, and several scholars of mythology believe that Hermes was once an epithet of Pan, before being spun off into his own divinity.
Annabeth is proud of herself for pickpocketing the god of thieves. Makes sense since in the books, her fatal flaw is hubris. Fortunately, Hermes let her pickpocket him.
During a flashback with Sally and Poseidon, she asks the god of the seas if he wants to talk to Percy. While Poseidon truly wants to, a distant roar of thunder is shortly heard after, signifying Zeus' wrath and monitoring of his fellow gods, which forces him to deny the offer.
In a departure from the books, shades in Asphodel slowly transform into trees. This brings to mind another tree with a human soul: Thalia's pine. Just as the Asphodel spirits are trapped in an eternal middle between Elysian Paradise and the Fields of Punishment, Thalia's spirit is somewhere in between life and death. As long as the tree lives, Thalia's soul will not go to the Underworld (where she certainly will not get a fair trial on account of being a forbidden child), but she can no longer participate in the world, while other people's lives happen around her. While this thankfully won't last, it would add a whole new layer to her decision to join the Hunters later on.
Poseidon speaking in a British accent despite the other Olympians having American accents. Britain is an island country with one of the most powerful navies in the world. Poseidon, as the god of the seas, probably would take aspects from a country closely tied to his domain.
On top of Colorblind Casting (and the general awesomeness of the late Lance Reddick), Zeus taking on the form of a black man does lean into one of his myths and characterizations. As told by Ovid, Zeus once disguised himself (and Hermes) as poor beggars to see how people would treat him. The couple who treated the two of them well were blessed; the people who rejected him were punished. Sacred Hospitality is one of Zeus' domains as a god and was taken seriously in Ancient Greece, and while beggars aren't quite common enough in the West to use that method, race relations are also a decent way to test mortals. It can even be seen as a mix of Ancient Greece's morals (guest rights are sacred) and the ideals of the modern West (that of racial equality), where Zeus tests both.
When he appears in Percy's dreams, Kronos carries around a lit lantern. Right before they are exposed in the Season 1 finale, what is the true Lightning Thief carrying? A lantern.
The Stinger where Gabe is revealed to have been divorced by Sally, locked out of the apartment, and accidentally petrifies himself with Medusa's head when he steals the package meant for Percy makes sense from a storytelling angle. In the books, Gabe was physically abusive to Sally for years, thus justifying her using the head on him. The series version is an Adaptational Wimp and thus didn't deserve such punishment, but was still a lazy, greedy, and repulsive person, so it's appropriate that instead he suffered a Karmic Death due to an vindictive act of pettiness backfiring on him.
It's likely Hephaestus knew Ares would send in proxies to retrieve his shield rather than come himself. So, he knew his trap would kill an innocent demigod who had nothing to do with their feud and was Just Following Orders.
