Daenerys Targaryen and her Court
Like in the TV show, Daenerys opened the Meerense fighting pits but instead of being attacked by The Sons of the Harpy, Drogon is attacked by an animal handler and Daenerys flies away on him as the pit fighters attack. Her story ends with the Dothraki finding her once again.
The main differences in Daenerys book storyline involves Barristan Selmy still being alive and as Tyrion and Jorah have not returned to Meereen yet, Sir Barristan takes over control of Meereen with Missandei to help, who is a young child in the book. She also has a personal bodyguard Eunuch called Strong Belwas assisting her.
After killing his father, Tyrion flees Kings Landing without Varys. He journeys to Volantis with Griff and Young Griff, a banished knight and a lost Targaryen descendant respectively. Jorah then captures him as the boat stops off in a town and they join with another dwarf called Penny on their way back to Daenerys. The three are eventually captured by slavers and taken to fight in the Meerenese fighting pits where Daenerys leaves on Drogon. The three then join a mercenary company after their Master dies.
Jaime's storyline starts to digress from the TV show after he returns home to Kings Landing. Instead of being sent to Dorne, Jaime is sent to the siege of Riverrun, where it is still loyal to the late King of the North Rob Stark and persuades Edmund Tully to yield the castle to him. He receives a letter from Cersei to assist her with the trial against her, but he instead burns the letter and instead goes searching for more Stark loyalists. He is then met by Brienne who tells him about a lightning of Sansa with the hound and the pair ride off to find her.
Cersei's storyline remains pretty much unchanged from the books.
The Stark Children
After fleeing the purple wedding, Littlefinger gives her the name Alayne Stone and takes her to The Vale where he weds her aunt, Lysa. Lysa attempts to kill Sansa by pushing her out the Moon Door but Littlefinger pushes her out first and then frames it on Lysa's singer Marillion. Unlike the TV show, Sansa does not travel to Winterfell to marry Ramsey Bolton, but instead her childhood friend Jeyne Poole is sent pretending to be Arya. Sansa remains at The Vale, recruiting to take back her childhood home. A later sighting of her near Riverrun spurns Brienne of Tarth and Jaime to go after her, although there is evidence this isn't her and she is still at The Vale.
Arya's TV storyline has more or less followed the book, apart from in the book Arya has a lot more names on her list of people to kill and therefore kills more regularly than the TV show.
Jon Snow's storyline is very much the same at the TV show, the book storyline has just ended with his stabbing by the Night's Watch.
Bran remains Beyond The Wall and his differences include a mysterious figure called Coldhands who helps the group on their journey and he instead dies as the group enter the cave of The Three Eyed Raven instead of Jojen, who remains alive.
Rickon has taken refuge on the island of Skagos whilst most of the world believes him to be dead. Davos Seaworth has just been sent to receive him.
Robb is also killed at the Red Wedding and his major storyline difference is his wife is called Jeyne and survives the Red Wedding instead of being killed.
Brienne of Tarth
After the Red Wedding, Brienne is sent out to search for the missing Sansa Stark with Podrick Payne. Unlike the TV show, she never encounters Arya on her journey and the pair instead run into the Brotherhood Without Banners who are now led by an undead Catelyn Stark. She judged Brienne and Pod as traitors and gives them a choice to either kill Jaime Lannister or hang. Brienne refuses to kill him so agrees to be hanged but after seeing Pod struggling agrees to do kill Jaime. She is later seen dragging Jaime away from his troops to find Sansa, but this is probably a ploy to kill him.
Stannis Baratheon, Melisandre and Davos Seaworth
Stannis is yet to attack Winterfell and remains buried in the snow nearby with Shireen and Selyse are still alive. Melisandre remained at the wall with Jon Snow and instead of burning Mance Rayder at the stake, she glamours someone else to look like him and sends Mance off to Winterfell to rescue who they believe to be Arya, but is actually Jeyne Poole, a child who used to be friends with Sansa. After Theon and 'Arya' escape Winterfell, Stannis sends them to the Wall to reunite with Jon.
Davos is sent away from Stannis to try and garner troops for his army. House Manderly refuses to support Stannis and decides to execute Davos but they instead execute a lookalike and secretly ally against the Lannisters. The Manderly's agree to help Stannis if Davos journeys to the island of Skagos and recovers Rickon Stark.
Myrcella is still sent to Dorne to achieve a political alliance and Oberyn is killed by The Mountain in Kings Landing but thats pretty much everything the same. Jaime and Bronn never travel to Dorne to get Myrcella as there is no evident threat on her life. The Sand Snakes, of which there are 8 instead of 3, seek revenge for their father's death but Doran has them arrested instead. Ellaria is given a major role in the TV series, but in the books remains isolated after Oberyn's death.
Arianne Martell, who doesn't yet exist in the TV show and is the brother of Myrcella's betrothed Trystane, seeks to gain power by putting Myrcella on the throne and causing a war with the Lannisters. She plans to smuggle Myrcella away from Dorne but her plan is foiled by Doran and she is thrown in jail. She then leaves Dorne to find out the truth about the supposed survival of Aegon Targaryen, the lost Targaryen descendant Tyrion encountered.
Arianne and Trystane's other sibling, Quentyn, heads to Meereen in an attempt to marry Daenerys Targaryen, who rejects his proposal. He hopes to be able to steal a dragon by taming it, but is instead burned alive.