There are two main principles tested in this problem: PARALLELISM and MODIFIERS.

PARALLELISM:
The conjunction "but" tells us that we need a parallel structure. In the original, we do not have parallelism:
Medicare... covers the cost of X, but not with Y.

We can eliminate A, which is not parallel. B and C both contain "of other hospital services," which is parallel. D and E change the structure entirely, so we no longer look for parallelism.

MODIFIERS:
Let's address D and E first. By changing the structure to "... home health care, which is unlike other nonhospital services..." we have changed the meaning drastically. We're no longer talking about "home health care" and "nonhospital services" as two comparable, parallel elements. The "which" makes it a MODIFIER - the "nonhospital services" are now describing the "home health care." This doesn't make much sense, and it drastically changes the meaning of the original.
Eliminate D and E.

In B and C, the difference is between "making" and "for which." In B, the present participle "making" is acting as an adverbial modifier - it's modifying the entire clause that came before it. In other words, it's saying that the fact that Medicare covers X but not Y is the thing that's making beneficiaries pay 20%. This may not be grammatically incorrect, but it fails the meaning test.

The idea of "beneficiaries paying 20%" should only be modifying nonhospital services. In C, the noun modifier "for which" correctly captures this meaning.