Thanks for the great question.
Correct: "Gorillas have more than one feature that distinguishes them from other primates."
Incorrect: "Gorillas have more than one feature that distinguish them from other primates."
The verb "distinguishes" here is in a clause that describes the singular "feature." What Gorillas "have" is "more than one" of something. The "something" here is a "feature." The "feature that distinguishes them from other primates" is the full description of this something, with "that distinguishes them from other primates" acting as an adjective clause, describing the singular "feature."
When you see a verb in an adjective clause, think: what noun is this verb describing? When a new subject is not introduced in the clause, the verb must agree with the noun described.
Hopefully helpful,
Josh