[highlander_UW]

But the question is, what's the difference between the adjective muslim and the adjective islamic

I would say that "muslim" refers to a person, "islamic" refers to something pertaining to Islam. An example would be "islamic" doctrine, or "islamic" thought. A muslim would follow islamic doctrine.

[Petronski]

A muslim person would follow islamic law. But would it be less accurate to say a muslim person would follow muslim law? Or perhaps just less elegant?

I don't know, nor frankly am I all that concerned with getting it right. But it is intellectually interesting.