How did it happen in the movie?
After his close encounter with the Witch King, Frodo wakes up in Rivendell, home of the Elf Lord Elrond and his daughter Arwen. In fact, Elrond is the first person he sees when he opens his eyes, followed by Gandalf and Sam. Gandalf, ever the mysterious but caring wizard, refuses to tell Frodo why he was delayed, but we get to see the events at Orthanc as a sort of flashback. After he is allowed to leave the bed, Frodo catches up with his friends and with his beloved uncle Bilbo, who chose to retire in Rivendell, writing his book.
While Elrond and Gandalf discuss the events that lead up to this moment and the gloomy future ahead, guests from all the free peoples of Middle Earth arrive in Rivendell. One of them is Boromir, a Man of Gondor, who is so startled by either finding the shards of Narsil or meeting a perfectly clean Strider, that he drops this heirloom.
After reassuring him that he is nothing like his forefather, Arwen gives the Evenstar, symbol of her immortality, to Aragorn.

How did it happen in the book?
The first chapter of Book 2 still has the Hobbit Frodo as main focus, and gives an account of Frodo’s awakening and a great feast where Frodo meets for the first time the lady Arwen, catches up with Glóin (one of the Dwarves who teamed up with Bilbo to reclaim the treasure of Erebor, as told is ‘The Hobbit’) and is finally reunited with Bilbo. We are also told the story of Eärendil and Elwing, by means of a song written by Bilbo and Strider, affectionately called the Dúnadan by the elder Hobbit.

This is indeed quite different from the events that take place between the ride to the Ford and the Council of Elrond in the movie. Let’s take a more detailed look at those:

Frodo’s Awakening
Though indeed Gandalf sits by Frodo’s side when he awakes, Elrond is not there. In fact, we don’t meet the Elf lord until later, at the feast. In stead of Gandalf thinking back to his imprisonment in and escape from Orthanc, we get the full tale of what happened to Frodo after he faced the Ringwraiths at the Ford of Bruinen. Later, after Frodo’s had a rest, Sam enters and leads him out to a party in Frodo’s honour.

Gandalf’s memory of the events in Orthanc is different from the account he gives of it in the books: he doesn’t send a moth to fetch Gwaihir, because the eagle is sent by Radagast the Brown.

The Feast
None of the events covered in the second half of this chapter were included in the movie, except for the reunion with Bilbo and his strange reaction to seeing the Ring (that peek-a-boo moment was incorporated in the scene where Bilbo hands Sting over to Frodo). In the book, Frodo first spends a rather long time eating and listening to the account of Glóin who provides us with some extra information on the city of Dale and Erebor, thus reminding us of the events in ‘The Hobbit’. This chapter also offers us the only glimpse of the lady Arwen until the end of ROTK.

Added scenes
– After Frodo meets up with Merry and Pippin, there’s a moment where he is talking to Bilbo about his own adventure being quite different from what he expected. There’s nothing like this in the book: when Frodo and Bilbo first meet again, after Bilbo gets freaky for a moment, they exchange news from the Shire.
– Later on, Sam tries to tell Frodo that it’s time to go home again. There’s nothing like this in the books, either.
– After recalling the events in Mount Doom, Gandalf suggests that Aragorn could reunite the different kingdoms of Men. Elrond responds with: ‘He turned from that path long time ago. He has chosen exile’. Nothing could be less true: all his life has it been Aragorn’s ambition to reclaim the throne of Arnor.
– Boromir doesn’t arrive in Rivendell until the next morning, just in time to join the council meeting, though that is not the purpose of his coming. There is no such moment between Aragorn and Boromir meeting each other as perfect strangers, and Boromir only sees the shards of Narsil when Aragorn shows them to him at the council: when the Fellowship hits the road a couple of months later, Narsil has already been reforged and renamed Andúril.
– There are also no scenes between Aragorn and Arwen in the books. Arwen is only mentioned twice: in the Many Meetings chapter and at the end of ROTK (6.VI: Many Partings).
Arwen had already chosen to be with Aragorn many years before, in the year 2980 of the third age to be precise, while Arwen was staying with her grandmother in Lothlórien. Aragorn gave her the ring of Barahir (which he does carry in the movie). She does have something like the pendant used in the movies, however she gives it to Frodo when the Hobbits leave Minas Tirith after the War: it is a symbol of her place on the ships that sail West, which she gives to Frodo. The name she gives it (‘Evenstar’) is the translation in Westron of the name given to her by her people: Undómiel (roughly meaning ‘star of twilight’).

Mistakes
– There’s a couple of moments where the balconies in Rivendell are apparantly Hobbit-sized. This was brought to our attention by Rivka, and you can see it clearly in this pic.
– TheBaggins brought to our attention the fact that because their is little to no time between Bilbo’s departure from Hobbiton and Frodo’s arrival in Rivendell, the sudden kick-in of the ageing process seems sort of unbelievable and doesn’t really make sense.
– When talking to Frodo, Bilbo says ‘I meant to go back… wander the paths of Mirkwood… visit Laketown… see the Lonely Mountain again. But age it seems has finally caught up with me.’ However, in the book Bilbo tells Frodo that ‘after a rest, I went on with the dwarves to Dale; my last journey.’
This change probably has to do with what was said above about the time lapse: if there was indeed hardly any time between Bilbo’s and Frodo’s respective departures, then Bilbo wouldn’t have had the time to travel to the Lonely Mountain and back.
– Boromir, ladies and gentleman, has black hair and grey eyes, much like Aragorn. Almost all the men in Tolkien’s books, who all come from the same forefathers, have dark hair. The only exception to this are the Rohirrim, who are almost all blondes.
– Boromir couldn’t have arrived on horseback, because he lost his horse on the journey from Minas Tirith to Rivendell and had to make the rest of it on foot.
– Legolas doesn’t ride saddle-less, this was noticed by Lozza.

Borrowed Lines
– Arwen’s ‘You are Isildur’s heir, not Isildur himself’ is originally used by Aragorn at the council of Elrond to pursuade Boromir of his sincirity.

Bookie Details
– This sequence kicks off with the exact same lines as the Many Meetings chapter in the book: ‘Where am I, and what is the time.’ – ‘In the House of Elrond, and it is ten o’clock in the morning. It is the morning of October the twenty-fourth, if you want to know.’
– When Sam enters the room, he takes hold of Frodo’s left hand and strokes it gently. This is a reference to the book, and one mentioned by Ian McKellen who thought that heterosexual actors would easily overlook this small token of affection and care. The reason why it is the left hand, is that this hand became numbed after Frodo was stabbed; a first sign of his changing into a wraith himself.
– There is a moment, before Frodo decides that it’s time to return to the Shire, when Sam is packing his bags and mumbles: ‘Now what have I forgotten?’ Rope, of course, as he will remember in the book when the Fellowship sets out from Rivendell.
– The book Bilbo has written is called ‘There and Back Again. A Hobbit’s Tale by Bilbo Baggins’. That’s not only the title which is used in the trilogy, it’s also the full title of ‘The Hobbit’, which is presented as Bilbo’s account of his journey with the Dwarves. When Frodo leafs through it, we can see a drawing of two swords, Sting and Glamdring, which is also in the original book.
– Elrond is described as having ‘hair (…) dark as the shadows of twilight, and upon it was set a circlet of silver’.
– One of the Dwarves that enters Rivendell in this sequence is an elderly looking fellow with a snow white forked beard and snow white hair: this is the aforementioned Glóin son of Gróin, father of Gimli. He doesn’t appear at the council, however.
– When standing on the bridge, Aragorn says about their first meeting: ‘I thought I had strayed into a dream’. As a matter of fact, Aragorn walked into Arwen in the gardens of Rivendell after she had spent a rather long time in Lórien, whilst singing the lay of Beren and Lúthien. Because she resembled Lúthien so much, he called out to her, as Beren had: ‘Tinúviel, Tinúviel’.
– The Evenstar is described as ‘a white gem like a star that lay upon her breast hanging upon a silver chain’.