Who is Háma in the movie?
When Aragorn, Legolas, Gimli and Gandalf arrive at Edoras, they are greeted before the doors of Meduseld by Rohirric guards. Their leader is Háma, a red-headed doorwarden with an overly cute accent. They are told to leave their weapons at the door, but Háma does show Gandalf respect by allowing him to keep his staff.
When Gandalf tries to ‘cure’ Théoden, Háma prevents Gamling from intervening. Afterwards, we see him round up the inhabitants of Edoras for the evacuation.
Háma dies during the Warg-attack.

Who was Háma in the book?
Háma was one of the doorwardens of King Théoden in Edoras. He receives Gandalf; Aragorn, Legolas and Gimli outside Meduseld, and – at Gríma‘s request – makes the company leave their weapons at the door. He is however pursuaded to allow Gandalf to keep his staff. This does not go down well with Théoden, who reduces his role to errand-runner.
Háma then fetches Éomer, who had been imprisoned in his own home, believing this to be his task. Théoden understands just how great Gríma’s influence over him was, and decides to ride out to battle. Since he has lost his son, Théodred, he apppoints Éomer as his heir. The problem then arises who is to lead the refugees to Dunharrow. Háma suggests Éowyn, and Théoden agrees.

Háma dies in the Battle of the Hornburg. He is buried before the gate, a place of special honour: In the midst of the field before the Hornburg two mounds were raised, and beneath them were laid all the Riders of the Mark who fell in defence, those of the East Dales upon one side, and those of the Westfold upon the other. I a grave alone under the shadow of the Hornburg lay Háma, captain of the King’s guard. He fell before the Gate. [3.VII.]

Who is Gamling in the movie?
Gamling is a member of Théoden’s household, and his guard. After Háma dies, he presents himself as Théoden’s second in command. He is present when the King puts on his armour, and doesn’t leave his side throughout the entire battle of Helm’s Deep.

Who was Gamling in the book?
Gamling the Old is the leader of a group of Rohirrim who defend Helm’s Dike during the battle of Helm’s Deep. He is introduced to us when he voices his concern about the soldiers left to defend the keep: …we have a thousand fit to fight on foot. But most of them have seen too many winters, as I have, or too few, as my son’s son here. [3.VII.]
Gamling is the one who discovers the Orcs have made a hole to crawl through, and with Gimli’s help tries to close it. He also translates the word of the Dunlendings for Aragorn, and gets stuck with Gimli in the caves by the end of the battle.

Differences between the books and the movies
In general one could say that Háma’s part was greatly downsized, while Gamling got a promotion. That Háma was important for Théoden can be derived from the following quote: …Théoden mourned for the loss of Háma, his captain, and cast the first earth upon his grave. ‘Great injury indeed has Saruman done to me and all this land,’ he said; ‘and I will remember it, when we meet.’ [3.VIII.]
Théoden does mention Háma’s death when he talks to Saruman.

Many things that Háma was supposed to say or do, were given to someone else:
– Háma is not berated for allowing Gandalf to keep his staff.
– Éomer was banished, not imprisoned, so Háma doesn’t free him.
– Éowyn travels to Helm’s Deep with the others, and the problem of who will keep the people save if Éomer and Théoden should die isn’t addressed until ‘The Return of the King’.
– Háma doesn’t make it to Helm’s Deep: he dies during the warg attack.

The only thing Movie Gamling and his book equivalent have in common, is their name.