I had the opportunity of doing some in depth courses on Christianity, and based on what I have learnt, these are the things that I found rather similar as compared to the Silmarillion.
First of all, as we all know, Tolkien described Eru as the creator of all things and that he was the one divine being that knew all and controlled all things. Needless to say, the Christian faith is based on one true God. This is not the only basis for arguing that his tale was based on the Christian faith however.
When Eru gave to each of the Ainur a portion of the flame imperishable, and subsequently Melkor searching the void in an attempt to find and obtain more of the flame, leading into Gandalf saying to the balrog that he is a servant of the secret fire at the bridge of Khazad-dum, it showed a striking similarity to the biblical interpretation of the Holy Spirit.
The Holy Spirit in the bible is described as several things. The Hebrew word used in most of the Old Testament, when refering to the Holy Spirit, is "Ruach", which means wind/breath. This also signifies life, as in the breath of life. When Eru created the Ainur, he gave to each part of the flame. Think of it as giving each of the Ainur the breath of life.
Now we talk about fire. In the story of Elijah, when Elijah offered a sacrifice to God in front of all the other pagan priests to demonstrate His power, God sent down a flame from heaven that consumed the offering. In the book of Acts in the New Testament, on the day of Pentecost when the Holy Spirit first came down and filled the people, it came as a tongue of flame that appeared on every person's head. This very same Holy Spirit that was called the breath of life earlier manifested later as a flame.
In the process of creation, the breath of life was needed. And Tolkien called it the flame imperishable. Now we can choose to read it as a force or power of some sort, since Melkor sought after it. But later we read that Melkor could not find in the void the flame because it was with Eru. This reads like the book of John, which says in the first chapter "In the beginning the Word was with God, and the Word was God". Now we know in that particular case, it was talking about Jesus. But we also know in the Christian faith, there is the concept of the Holy Trinity, which means that the Father, Son and the Holy Spirit makes up what is called the "Godhead", or simply put, God.
Furthermore, we cannot consider it simply as a force because Gandalf was a servant of the fire, and it would be incorrect to say that someone was a servant of a force (being an energy that does not have a will). Hence it would be right to say that Eru WAS the flame imperishable... which would make it similar to the Christian concept of God and the Holy Spirit are one and the same. It would also show that when the flame was given to empower the Ainur, it is exactly the same principle as when Jesus gave each of the believers on the day of Pentecost the Holy Spirit to empower them to do His work.
It is also mentioned that Eru chastised Melkor when he sought to make his own music, saying that whatever other tune he made which he thought was separate from Eru was in fact part of the plan of Eru, since Eru was the creator of all music. This is exactly the same as the Christian belief that all things are according to the will and plan of God.
Tolkien's wonderful use of music as the key in creation also displayed Christian beliefs and ideology. In the beginning, the main activity that the Ainur did was making music. It is later mentioned at the end of Ainulindale that no greater music shall be made till the end, when all creations of Eru, including the Children of Illuvatar shall sing the greatest music ever. In Christianity, heavenly hosts of God are constantly making music to praise God. God loves and desires the making of music for His praise, seen in many instances especially in the case of King David, who managed to have the presence of God constantly on the ark of covenant manifested as a blue flame as a result of praise and worship through singing 24 hours a day. It is also believed by Christians that at the end of days, believers will all join in the great chorus in heaven to praise God.
In a certain way, Tolkien calling the elves his children is similar to God calling the Jews his children in the beginning, later accepting even Gentiles if they believe. In Christianity, believers will have everlasting life, likened to the immortality of the elves.
The characteristics of Melkor is almost the exact same interpretation of Lucifer. The main characteristic (and fault) is the same. Pride.
The Hall of Mandos is basically an interpretation of the judgement of God, but then again since Tolkien was a Catholic, it most likely was derived from the concept of Purgutory.
It is important to note however, that Tolkien was attempting to create a story worthy of being called a myth that belonged to the English culture. Hence it was meant to be a purely fictional piece of work. Tolkien was not attempting to write Christian literature. He was writing a story. But his strong beliefs and convictions in his faith has undoubtedly caused him to create a world that had such strong similarities, and to write about divine beings as he envisioned them to be.
Hope I wasn't too preachy and that anyone reading this would be able to understand it. Trying to explain it at 1.30 in the morning is challenging indeed!