Adding up to the previous answer: if we keep to LotR and The Hobbit, the Seven Virtues of Catholicism are predominant in all Tolkien's work. The most vivid examples of the virtues I can think of, in a cetain disorder:
- Charity, in a self-sacrifice and generosity sense. Frodo accepts to risk everything so as to cast the One Ring into the fire. No one asked explicitly for him to continue on, and there were countless other great figures, Elves, Men and Dwarves alike, at the Council of Elrond. When Frodo decides to take upon the burden, Elrond responds to him:
"I do not lay it on you. But if you take it freely, I will say that your choice is right, and though all the mighty elf-friends of old, Hador, and Hurin and Turin, and Beren himself were assembled together, your seat should be among them."
- Patience, from which comes forgivness, mercy. What would have been the fate of the world would not have Bilbo spared Gollum on his way out of the Misty Mountains?
Frodo: "It's pity Bilbo didn't kill him when he had a chance!"
Gandalf: "Pity? It was pity that stayed Bilbo's hand. Many that live deserve death. And some that died deserve life. Can you give it to them Frodo? Do not be too eager to deal out death in judgment. Even the very wise cannot see all ends. My heart tells me that Gollum has some part to play yet, for good or ill, before this is over. The pity of Bilbo may rule the fate of many."
- Kindness, which radiates from positive thinking and outlooks. Probably the best example is Frodo's relationship with Gollum during his journey trying to enter the dark land of Mordor: Gollum almost gives up to his remorse before they enter Shelob's lair. Also, the cheerful talking is what keeps Frodo and Sam going even in the darkest of the places, like Cirith Ungol stairs:
'It's saying a lot too much,' said Frodo, and he laughed, a long clear
laugh from his heart. Such a sound had not been heard in those places
since Sauron came to Middle-earth. [...] 'Why, Sam,' [Frodo] said, 'to hear you somehow makes me as merry as if the story was already written.'
- Humility. Although knowing his glorious lineage, Aragorn stays humble throughout his entire life. After the siege of Gondor and the great battle of Pelennor's field, he stays at the entrance of Gondor and continues to go among ordinary people and heal best as he can.
[...] men came and prayed that he would heal their kinsmen or their friends whose lives were in peril through hurt or wound, or who lay under the Black Shadow. And Aragorn arose and went out, and he sent for the sons of Elrond, and together they laboured far into the night. [...] And when he could labour no more, he cast his cloak about him, and slipped out of the City, and went to his tent just ere dawn and slept for a little.
- Temperance from which comes restraint, honor, justice. Faramir treats Frodo fairly, and later by extent Gollum too, as he swears he's no orc spy but their guide. When lecturing Sam, after the Ithillien ambush:
'Patience!' said Faramir, but without anger. 'Do not speak before your master, whose wit is greater than yours. And I do not need any to teach me of our peril. Even so, I spare a brief time, in order to judge justly in a hard matter. Were I as hasty as you, I might have slain you long ago.'
- Diligence, which gives persistence, fortitude. Examples are plenty, although the most striking one is the pursuit by Aragorn, Legolas and Gimli of the two captured hobbits.
- Chastity. Sexuality is never really tackled in Tolkien's work. However, Aragorn is promised Arwen since he was young, relatively speaking, if he's deemed worthy by Elrond, and for all we know, Aragorn may have very well waited this long for his love.
As to the difference with other Catholic branches, such as Anglicism or Protestantism, can be mainly viewed with the Pope's authority.
Gandalf, well-known magician in most part of the world, can be compared to an archbishop. After his death, although successfully defeating the Balrog in Moria, he becomes the Pope: head of the White Council by Ilúvatar's decree. Everyone listens to his advice, those who don't usually get somewhat of a bad fortune (Boromir and Denethor come first into mind). Like in historical events, he even crowns Aragorn king.