What's the difference between the names Eru & Ilúvatar?

Question

If my Silmarillion memory serves, both are names of the ultimate creator. What's the difference? When should one used over the other?

Answer 1

'Ilúvatar was the first beginning, and beyond that no wisdom of the Valar or of Eldar or of Men can go.'

'Who was Ilúvatar?' asked Eriol. 'Was he of the Gods?'

'Nay,' said Rúmil, 'that he was not, for he made them. Ilúvatar is the Lord for Always who dwells beyond the world; who made it and is not of it nor in it, but loves it.'

― The Book of Lost Tales Part One, "The Music of the Ainur"

• Eru is a Quenya name meaning "He that is Alone".

• Ilúvatar is Quenya for "the Father of All", more commonly referred to as Eru.

• The name Ilúvatar is a compound of two words, ilu and ilúvë "universe" and atar "father."

• In the earlier versions Ilúvatar was the main name of God used. Another name was Ainatar "Father of Gods" — the word Eru first appeared in The Annals of Aman.

When you combine Eru and Ilúvatar, it means "He that is alone the father of all." Neat, isn't it?

Further information can be found in here.

Answer 2

Ilúvatar means "Father of All" in Quenya, while Eru means "The One", or "He that is Alone".

According to Wikipedia, both names are used in the same way, even appearing paired (Eru Ilúvatar) in several occasions. Don't seem to be a "rule of thumb" about when one should be used over the other.

Answer 3

The meaning of Eru is similar to one interpretation of the meaning of Yahweh, which is "he who is". Perhaps that's not surprising, considering that in Tolkien's mind, Eru and Yahweh are, in some sense, one and the same.

Ilúvatar can probably be looked at as an epithet. With its meaning "father of all", it is akin to Odin's epithet "all-father". The question of which name is more widely used is unclear to me — Tolkien Gateway says "usually called Eru", while Encyclopedia of Arda says Ilúvatar is his "name among the elves" and that Eru is "more commonly known in Middle-earth as Ilúvatar." My own guess is that this is correct, and that Ilúvatar is a term of polite reverence, similar to the elves' name Elbereth (star-queen) for the Ainu Varda.

Fëanor is known to have sworn his oath by both names, translated as "our word hear thou, Eru Allfather".