The Greek gods play a central role in all the events as they are depicted in The Iliad. Paris takes Helen from her home only after she is promised to him by Aphrodite (which was a separate story not directly told in the Iliad). There had been a quarrel between the Goddesses Aphrodite, Hera and Athena about who was the fairest. They asked the God King Zeus to choose, but knowing that picking one would only incur the anger of the others, he appointed the mortal man Paris to pick the Fairest of the Goddesses. To make his choice, Paris had to listen to their bribes and pick the best. Hera offered him land; Athena offered him skill and wisdom; and Aphrodite offered him love. Being a ladies' man with little ambition, Paris chose Aphrodite's bribe to have as his own the most beautiful woman in the world - Helen.
Because of his decision, Aphrodite assists Paris and the Trojans throughout the war, as does her lover Ares, and Apollo. Conversely, Hera and Athena are on the side of the Greeks.
There is a scene in the Iliad where the gods themselves do battle. Athena bests Ares and Aphrodite, while Hermes surrenders before Leto because she's a paramour of Zeus. Apollo refuses a challenge from Poseidon, and when Artemis criticizes her brother, Hera slaps her down.
In the book, the war takes about ten years, compared to several days in the movie.
In the Iliad, Agamemnon is not the butcher and warmonger portrayed in the movie. An unrelated story goes that many Greek kings vied for the hand of Helen, and Odysseus saved them from warring against each other by declaring that Helen herself should choose her husband, and all Greek kings will come to the aid of her husband if anything happens to her. Helen chose Menelaus, and when she is seduced by Paris, all the Greek kings sail to Troy in honor of their pledge.
Hector knew he was fighting Patroclus instead of Achilles, in the Iliad; Apollo reveals him, and also helps Hector defeat him. Hector claims Achilles' bronze armor, which Patroclus was wearing, for his own. Thetis, Achilles' mother, enlists Hephaestus, blacksmith of the gods, to forge new armor for Achilles, including a magnificent shield inlaid with gold.
Rather than approaching the city alone, Achilles leads the Greek army to the gates of Troy in a great rout before tracking down Hector. Upon being faced with Achilles, Hector flees three times around the city and only stops to fight when Athena tricks him. Their battle, unlike in the movie, is extremely short in the story. Hector ducks Achilles' thrown spear, while Hector's spear is blocked by Achilles' shield. Athena returns Achilles' spear to him, and he kills Hector with a thrust through the neck just as Hector closes with his sword. Apollo helps Paris kill Achilles later on.
In the story, Menelaus and Agamemnon survive the war. Menelaus gets Helen back, and they end up raising a family. Agamemnon eventually returns home, only to be killed by his wife and her lover. (In an unrelated story, Agamemnon's son, Orestes, later kills Helen, holding her partly responsible for Agamemnon's death.)
Odysseus has to endure the events detailed in The Odyssey before finally making it home. Just prior to returning, he stops and visits Menelaus and Helen at their home.
Prominent warriors on both sides- Aeneas for the Trojans and Diomedes for the Greeks- are left out of the movie. Achilles' son, Neoptolemus, is also left out.
It isn't Paris who warns against the Trojan Horse, but the Trojan prophetess, Cassandra. However, nobody believes her because of a curse placed on her by Apollo when she spurned his love.
Paris dies before Troy is sacked, killed by the legendary Greek archer, Philoctetes. Ajax does not die in battle. He commits suicide.
The Iliad ends with the funeral of Hector. The Trojan Horse, fall of Troy, and other post-Iliad events are told in other stories.