How did it happen in the movie?
Frodo wakes up as Faramir walks beside him and asks Frodo to come with him. They go out and Faramir asks Frodo to look down where he sees Gollum diving in the pool fishing. Faramir says entering the Forbidden Pool bears the penalty of death and shows Frodo his men ready to shoot Gollum. Faramir says: “They wait for my command. Shall I shoot?” Frodo listens to Gollum singing some kind of song and after a few seconds says not to shoot because he’s their guide, and asks permission to go to him. Frodo goes down to persuade the fish eating Gollum out of the pool.
Frodo: Sméagol. Master is here. Come, Sméagol. Trust master. Come.
Smeagol: We must go now?
Frodo: Sméagol, you must trust master. Follow me. Come on. Come. Come Sméagol. Nice Sméagol. That’s it. Come on.

Gollum follows Frodo and when Faramir’s men catch him, he cries in disbelief “Master!” Inside the cave Faramir’s men kick and hit Gollum until Faramir stops it and asks where Gollum is taking Frodo and Sam. Gollum doesn’t answer and lying in the corner Gollum talks with Smeagol and they reveal that the hobbits have stolen something from them. Faramir asks what they have stolen and Gollum cries “My Precious!”

In the other room Sam asks Frodo to use the ring and go but Frodo says he can’t because Sam was right, the Ring is taking him. Faramir comes and uses the tip of his sword to take out the Ring so that he can see it. He then says “So this is the answer to all the riddles. Here in the Wild I have you, two Halflings and a host of men at my call and the Ring of Power within my grasp.
A chance for Faramir, captain of Gondor, to show his quality.”
Sam begs Faramir to let them go because they have to destroy the Ring but Faramir who is obviously tempted by the Ring, says the Ring will go to Gondor.

How did it happen in the book?
Frodo wakes up when Faramir bends over him and asks Frodo to come with him. Sam wakes up after they have gone and follows them outside the cave. Faramir asks Frodo to look down and he sees something diving into the pool. Faramir and Anborn wonder what the creature is and their men are waiting for Faramir’s command to shoot him. Turning to Frodo he asks “Shall we shoot?” Frodo thinks for a moment, says no and Faramir asks if Frodo knows then what this thing is. Frodo explains that Gollum probably doesn’t know that Men live in this place and that he’s lured here because he has born the Ring for many years. They see Gollum fishing and Faramir still asks why he should be saved. Frodo says he’s their guide and asks if he can try to get him out of the pool.

Gollum is hissing in the pool cursing the hobbits and Men. Frodo goes to the pool and calls for Gollum but he wants to finish his fish first. Frodo desperately says “Smeagol! Precious will be angry. I shall take Precious, and I shall say: make him swallow the bones and choke. Never taste fish again. Come, Precious is waiting!” Now Gollum crawls to him and wants to continue the journey immediately but Frodo just asks him to trust him: “Yes, we’ll go soon. But not at once. I will go with you as I promised. I promise again. But not now. You are not safe yet. I will save you, but you must trust me.”
Frodo feels bad for doing this to Gollum and Gollum is very doubtful and feels that something’s wrong. When Anborn gets Gollum he cries: “Masster, masster! Wicked! Tricksy! False!” They take Gollum to Faramir who questions him about his business and what he knows about the place where he is. After a while Faramir is satisfied and let’s Frodo decide of Gollum. Faramir lets Frodo go free declaring that Frodo is free to walk in Gondor and that Gollum is under doom of death except when walking with Frodo. Faramir asks where Gollum is leading Frodo and Sam and when he hears he’s leading them to Cirith Ungol Faramir warns Frodo many times of this path and that Gollum is no to be trusted. Then he leaves Frodo and Sam alone resting for a while.

Changes
– In the movie the whole atmosphere of the scene is different because straight from the beginning Faramir was suspicious towards Frodo and Sam so he isn’t as kind towards them as in the book. Also when Faramir finds out about the Ring and sees it, he’s tempted by it and doesn’t let Frodo and Sam go but says that the Ring goes to Gondor. The writers say this was all about getting a proper ending for the journey of Frodo and Sam in TTT. Their journey to where? To what? And the answer was Faramir who had to become an obstacle for Frodo and Sam. They also wanted Faramir’s character to go through real development and they thought that if Faramir isn’t tempted by the Ring at all it would have diminished the power of the Ring now after they had tried to prove how bad and treacherous the Ring was.
– In the book Sam is at the pool with Frodo and Faramir.
– The movie doesn’t name any of Faramir’s men but in the book Anborn is named and he is there with Faramir actively.
– In the book wine is offered to Frodo and Sam and Faramir’s men don’t kick and hit Gollum but only question him. After that Faramir actually says: “Take this creature away, Anborn. Treat him gently but watch him.”

Bookie Details
– The view at the background outside the cave where Frodo and Faramir look down to the pool is straight from the book: “Far off in the West the full moon was sinking, round and white. Pale mists shimmered in the great vale below: a wide gulf of silver fume, beneath which rolled the cool night-waters of the Anduin. A black darkness loomed beyond, and it glinted, here and there, cold, sharp, remote white as the teeth of ghosts, the peaks of Ered Nimrais, the White Mountains of the Realm of Gondor, tipped with everlasting snow.” [4.VI.] You can see it here.