Moving straight from the Ferry to Bree introduces an awkward inconsistency, because it was during their adventures in the Barrow-downs that the Hobbits acquired their weapons. To fill this gap, we have Aragorn producing a satchel of short swords on Amon Sûl, though the film doesn't attempt to explain how he got his hands on this convenient selection of hobbit-sized weaponry.
Arwen's role has been greatly expanded in the film. In the book, Aragorn and the Hobbits are aided by a golden-haired Elf named Glorfindel, not by Arwen, and the flood that saves Frodo from the Black Riders is the work of Elrond and Gandalf.
The film tells us that Aragorn has renounced his Kingship, but in the book he does no such thing. He is acknowledged leader of his people, the Northern Dúnedain, and by virtue of being Isildur's Heir has the right to claim the throne of Gondor. That he hasn't done so yet doesn't mean he doesn't intend to.
In the book, there are not two but three Wizards; the third, Radagast the Brown, has been removed from the film. He plays only a small part in the story, but he's important in that he unknowingly arranges for Gandalf's escape from Orthanc by sending the Eagle Gwaihir. With no Radagast in the film, Gandalf has to arrange his own escape, by sending a message to Gwaihir by moth.
In the book, when the Fellowship attempt the Redhorn Pass, they are most likely beaten back by the ill will of Caradhras itself, or just possibly through the power of Sauron; the film shows Saruman causing their difficulties instead.
In the book, Saruman bred Men and Orcs to create creatures known as Half-orcs or Goblin-men, and this has led some to associate him with the origins of the Uruk-hai too. The original Uruks were full Orcs of a particularly powerful and deadly kind, and originated in Mordor about five hundred years before the story begins. The film chooses to make Saruman their creator, but it isn't clear whether this is intentional departure from the book, or a simple mistake.
Lurtz, the Uruk chieftain who shoots Boromir and is in turn slain by Aragorn, is entirely an invention of the film-makers. No such character appears anywhere in the book. He does seem to be quite compatible with the story, though. In fact, if the film version of the The Two Towers follows the book of the same name, we'll see his Orc-band arguing and fighting among themselves, which would make sense if they'd lost their leader in battle.
Facts:
Simplifying the War of the Last Alliance sees the entire seven years of the Siege of Barad-dûr removed from the story, but there is a nod of acknowledgement of this in the film. In Tolkien's chronology, Sauron fell in the year 3441 of the Second Age, but in the movie, we see Isildur writing his Scroll about the Ring in 3434, which would have been the correct year if the Siege had never taken place.
Though Elendil isn't named in the battle scene, he does briefly appear - he's the warrior next to Isildur, killed by Sauron, whose sword Narsil Isildur snatches up. His name does appear once in the movie, though: near the end, as Aragorn leaps on a band of Orcs, he shouts his battle-cry 'Elendil!'.
When Gandalf visits Bilbo in Bag End, there's a framed map on the table, which he picks up and examines. In fact this is Thrór's Map from The Hobbit, that Gandalf had himself recovered from Sauron's dungeons 151 years earlier, making Bilbo's first adventure possible.
When Bilbo offers Gandalf a glass of 1296 Old Winyards, he mentions that it's 'almost as old as I am'. Bilbo was born in 1290, so the wine was laid down (by his father Bungo, as the book tells us) when he was just six years old.
Frodo is much younger in the movie than the book. According to Tolkien, he was born in the year III 2968, and left the Shire in III 3018, making him exactly fifty years old. It should be said, though, that hobbits age more slowly than humans (and Frodo's ageing was also slowed by his ownership of the Ring) so this isn't really a drastic departure (Aragorn, who comes from a line of very long-lived and slowly-ageing Men, is actually eighty-seven years old!)
As Frodo and Sam are leaving the Shire, just before they meet Merry and Pippin, we see them walking through a field of corn. When Tolkien writes about 'corn', though, he uses the word in its British sense, meaning a crop like wheat. The corn we see on the screen, though, is maize, a plant native to the Americas that couldn't possibly have existed in Middle-earth (which represents the lands that now lie east of the Atlantic Ocean). There's a similar slip later in the movie where Merry and Pippin cook tomatoes - there were no tomatoes in Middle-earth, either, for just the same reason.
At least three of the chapter titles from the book have made it into the film, as part of the dialogue. Look out for 'a long-expected party' (the title of chapter I 1), 'a short cut to mushrooms' (chapter I 4) and 'the bridge of Khazad-dûm (chapter II 5). Gandalf also uses the phrase 'riddles in the dark', which is the title of the chapter in The Hobbit that sees Bilbo acquire the Ring.
Bree is a town shared by Men and Hobbits. Though this isn't perhaps obvious in the movie, there is a neat touch in that the town's gatekeeper has two spy-holes to look through, one at Man-height, and one at Hobbit-height.
Shortly after Frodo is wounded on Amon Sûl, there's a scene where Aragorn sends Sam to find athelas for Frodo's wound, just before we meet Arwen for the first time. In the background, we get a glimpse of huge stone figures - the three trolls that Gandalf tricked into turning to stone in The Hobbit, and whose story Bilbo briefly explains to a group of hobbit-children at his Birthday Party.
There's at least one line of Elvish in the film that isn't subtitled - Arwen's 'spell' that raises a flood to sweep away the Black Riders. She says 'Nîn o Chithaeglir lasto beth daer; rimmo nín Bruinen dan in Ulaer', which means roughly 'Waters of the Hithaeglir, hear the word of power, rush, waters of Bruinen, against the Ring-wraiths'. If you're interested in the Elvish used in the movie, there's a particularly useful resource at 'Tolkien's Languages in the FotR Movie'.
In Rivendell, Arwen gives Aragorn a piece of jewellery. There's no such event in the book, but much later in the story, she does give something similar, not to Aragorn, but to Frodo: 'And she took a white gem like a star that lay upon her breast hanging upon a silver chain, and she set the chain about Frodo's neck.' (The Return of the King VI 6). Some of the movie merchandising suggests that this has jewel has acquired the name 'Evenstar' (in the book, 'Evenstar' is Arwen's surname, a translation of Elvish Undómiel).
It's interesting that Saruman seems to stumble over the name 'Khazad-dûm'. Tolkien actually records the fact that Elvish-speakers had trouble pronouncing Dwarvish names, so it's just possible that the film-makers (who have taken enormous trouble to use correct pronunciations) included this touch intentionally. Actually, he would probably have just used the Elvish name for the place, Hadhodrond.
Among the travellers arriving for the Council of Elrond, we catch a fleeting glance of an old, white-haired Dwarf with Gimli. That's his father, Glóin, who travelled with Bilbo in The Hobbit.
The Argonath, the huge statues over the river that the Fellowship row beneath, are of Isildur himself (on the right, with the beard) and his brother Anárion.
In the book, Saruman's Orcs wear helmets marked with an 'S'-rune (). In the movie, they're marked instead by a white hand-print across their forehead. Actually, this isn't too inconsistent with the original story, because Saruman's symbol is elsewhere described as the White Hand.
In Lórien, Boromir asks Aragorn if he has ever seen his home, the White City of Minas Tirith. Aragorn's reply, that he has, long ago, is quite an understatement. In fact, he served there as a captain of Gondor's armies, under the alias 'Thorongil', and was personally acquainted with Boromir's grandfather Ecthelion II. The 'Tower of Ecthelion' that Boromir mentions in the same scene, though, isn't connected to his grandfather. It was built by his ancestor Ecthelion I some centuries earlier.
And here ends the Changes and Facts to "The Fellowship of the Ring".