congratulatory letter vs congratulations letter

congratulatory letter and congratulations letter both are nouns.

Word	Noun	Adjective	Verb	Adverb
congratulatory letter	Yes	No	No	No
congratulations letter	Yes	No	No	No

As nouns, congratulatory letter and congratulations letter are synonyms defined as:

congratulatory letter and congratulations letter: an official letter in which one's achievements are recognized and honored.

congratulatory letter (noun)	congratulations letter (noun)
an official letter in which one's achievements are recognized and honored.	an official letter in which one's achievements are recognized and honored.