These two nouns are completely unrelated—and Acclamate is not a word. Let's take a closer look at acclimation vs acclamation.
The difference should be easy to remember because acclimation is based on the noun, climate, while acclamation is based on the noun and verb, claim.
But don't let that confuse you over the word, Acclaim, which means “praise.” There is a reason the following two nouns are used distinctly: acclaim and acclamation.
Acclaim can only be a mass noun; it can only refer to praise as a substance:
Correct: She received much acclaim/acclamation for her theory.
Incorrect: She received *many acclaims for her theory.
In contrast, the noun acclamation can refer to either mass acclamation or an instance of praise:
Correct: She received much acclamation for her theory.
Acceptable: She received many acclamations for her performance.
Some grammar-checkers do not accept acclamation as a countable noun; however, it is commonly used in this way and acceptable.