[Scott]

Hello Lost One, and welcome to the Downs, if it hasn't already been said.
While your question is certainly a valid one, your answer lies really only in observing how languages evolve in the first place. From what I can gather, languages tend to evolve most often due to slang usage, due to outside innterractions (with other languages and peoples), and due to the speaker's environments.
You brought up the fact that since elves were immortal, their language shouldn't differ. If you look just at the different vernacular that has passed through the lips of American tongues within the last say fifty years, there are a plethora of slight differences and incongruities among what's said, even by those people who have lived that whole time. for example, the term groovy was one of my father's favorites, but almost noone uses that term on a regular basis today. Although I'm not suggesting that the elves would have used words sch as groovy, hip, or cool, I don't think it's hard to say that it was possible for them to have popular colloquial sayings that changeed from time to time.
You also mentioned that the divergences have a bit to do with the separation of the different groups of the elves. While separated into their different areas in Arda, the elves came into contact with other races and peoples of different languages than their own. The interraction of elves with men, dwarves, etc. could have perhaps caused shifts in the languages, if not even additions. For an example in our own world, French phrases such as trés chic can be found in most good English dictionaries.
The environments that the different groups of elves were in would also have an effect on their languages. Elves such as the Teleri would need words that would suit best their maritime environment whereas Sindarin elves would be more likely to form expressions of their woodland environmnets.
I hope my reply wasn't too wordy, or drawn out and that it helped answer at least some of your questions. And please, feel free to criticize it; I'm no philologist or etymologist, only a student.

~Scott