There are too many differences between the movies and the book to list, even over the course of the next year. These include things as minor as characters’ lines being swapped and major things like entire sequences and people being cut out. Some of these are for time’s sake, as a movie including every line and event in the book would be far too long for most audiences to enjoy. However, other alterations are made for artistic purposes or other story-telling reasons.
For example, in both the book and movie, Gandalf is captured by Saruman and placed on top of Orthanc, the tower he lives in. In the movie, you see Gandalf “communicate” with a moth-like creature and later a great eagle comes and rescues him. The sequence in the book plays out a little differently.
At this point in the story, Saruman’s treachery has not yet been revealed. Gandalf, on his way back to the Shire to warn Frodo of what he’s learned of the Ring, is met by Radagast the Brown on the road to Bree. Radagast informs Gandalf that Saruman offers his aid against the Nazgúl, but Gandalf must go there at once. Distressed, Gandalf leaves a message at Bree for Frodo and goes immediately to Isengard. Before he and Radagast part, Gandalf tells him to send messages to all friendly birds and beasts he can, telling them to keep an eye out for strange activity and to report to him what they can. Radagast agrees and they part ways.
Time moves forward and Gandalf is held prisoner by Saruman on the roof of Orthanc with no means of escape. Fortunately, Radagast was not under the influence of Saruman and did not know of his treachery for he took the advice of Gandalf and informed every beast and bird he could of what was happening. Due to that, Gwaihir, the Lord of the Eagles, came to Orthanc to deliver messages to Gandalf and found him on the pinnacle. From there, he bore Gandalf to Rohan the rest of the story plays out.
Peter Jackson’s reasoning on not including Radagast here is not definitely known but the result of the choices he made are shown throughout the trilogy. Gandalf’s encounter with the moth in The Fellowship of the Ring is seen again in The Return of the King at the battle in front of the Black Gate when the Eagles appear again.
[Credit to Jack Moore for proofreading and corrections.]