"Elven" is an (optional) substitute for "elf" as the first element of a compound word
As Valorum mentions in a comment, in general "elven" seems to be used in contexts where the noun "elf" would also be possible. This seems to be because "elven" was originally a noun like elf, not an adjective like elvish. The Oxford English Dictionary says the -en of elven is not in origin an adjectival suffix, but actually comes from a Germanic feminine noun suffix. Despite this, the word came to be used as a noun for an elf of either gender. This use is now obsolete, but it seems to have been retained in poetry longer than elsewhere. Tolkien was surely aware of all of this information, being a philologist and English professor.
The use of elven in words like Elvenking or elven-tongue may be comparable to Tolkien's use of dwarrow (an alternative form of the noun dwarf) in the compound word Dwarrowdelf: he does not use either "elven" or "dwarrow" as an independent noun. However, Tolkien does sometimes use the noun "elf" attributively in compounds, such as "elf-cake" or "Elf-country"; I don't know if there is any significance to using "elf" vs. "elven" in this sort of context. Perhaps "elven", being archaic, creates more of a sense of dignity.
I don't know of any case where "elven" is unambiguously an adjective in Tolkien's work; I would appreciate any examples if someone is aware of them. It certainly seems to have an adjective-like meaning in some expressions, but as far as I know it is always used attributively before a noun, and this is a position where nouns can occur as well as adjectives (for example, one can say "orc-king" or "orc tongue" rather than "orcish king" or "orcish tongue"); in fact, "elven" often occurs as the first element of hyphenated or closed compound words, and I think noun-noun compounds are more common in English than adjective-noun compounds (consider the compound words "Entmoot", "ent-house", "ent-stride", "ent-maiden" used in Tolkien's work).
Note that in the segment of letter 138 quoted by Valorum,
But the printing is very good, as it ought to be from an almost faultless copy; except that the impertinent compositors have taken it upon themselves to correct, as they suppose, my spelling and grammar: altering throughout dwarves to dwarfs; elvish to elfish; further to farther; and worst of all, elven- to elfin. I let off my irritation in a snorter to A. and U. which produced a grovel.
"elvish" is written as an independent word, but "elven-" is written with a hyphen after it. I am copying this from Valorum's comment; I don't have direct access to the letter so I can't absolutely verify that this detail was in the original. But if it is present there, it seems somewhat significant to me.
I thought of another issue this brings up: does "dwarven" exist, either as a noun or as an adjective? If this site can be trusted (http://lotrproject.com/statistics/books/keywordsearch) then in fact there are no examples of "dwarven" in Tolkien's work. This contrasts with 5 examples of "dwarvish" in The Hobbit. The word "dwarven" definitely shows up as an adjective in more recent fantasy that is derivative of Tolkien's work, but it doesn't seem to have the same deep historical roots as elven (dwarven is not yet in the Oxford English Dictionary, in fact!).
"Elvish" is used as an adjective, and as a noun referring to the language
Elvish is generally an adjective, and is used in circumstances that require a word that is unambiguously an adjective. It has the usual broad range of meaning of an "-ish" adjective, and can mean either "of an elf/elves" or "like an elf/elves". It is also used as a noun meaning "the language(s) of the Elves", just like "Entish" is the language of the Ents.