I've asked this very question in the past and my research led me to the following definition which (surprisingly) differs from every other answer here so far:
～となる expresses a discrete change, while ～になる can express either a discrete or a continuous change.
You can feasibly use ～になる for everything, since it covers all cases, but in cases where you want to use the most suitable grammatical phrasing (such as in formal situations), you often see this distinction made.
その島は去年、無人島となった。 (～となる because the change from "inhabited" to "uninhabited" is discrete and happens the instant the last person leaves.)
デビューした後、彼はだんだん人気者になっていった。 (～になる because popularity changes continuously in unmeasurable steps.)
スカイツリーは現在、日本一高い建造物となっている。 (～となる because the title of "tallest building in Japan" applies to exactly one building, and a building cannot gradually become the tallest. It either is or it isn't.)
季節はこれから夏になっていく。 (～になる because the change in seasons is a gradual, progressive one.)
電線事故のため、新山口駅は終点となります。 (～となる because the title of "terminal station" only belongs to one station, and a station can't hold it partially.)
Your example in the question illustrates a discrete change in state (or more precisely, a discrete difference between 現金 and all other forms of payment), so both ～となる and ～になる are acceptable. However, since ～となる is "built" for such situations as these, it sounds more "proper" to use ～となる.
I suspect this rule also applies to the ～とする and ～にする forms, but I would have to do more research to confirm this.