I know students that have attended both. In a broad way, I think a student who was happy at one could also be happy at the other.
Aside from one being public and the other private, and $, I’d say the next most significant difference is location. GC is in a large city, and while not immediately downtown (like UNCG) it is close enough to have full access to all a city that size can provide. Meanwhile, UNCA is in a smaller city, and Asheville will feel a lot different than Greensboro.
Also, while GC has about half the enrollment of UNCA, GC has approximately 250 more African-American students than UNCA. If you think your student would be fine on either campus, there are no concerns either way. However, if being on a more diverse (or less diverse) campus is important, that is something else to consider.
If your student is the sort that would spend 95% of his time on campus, the differences in location might be minimized. However, if your student might spend a large portion of his free time off campus, he should investigate and see whether he would prefer the attractions of Greensboro or those of Asheville. The cities each have their great points, but there are stark differences in several ways.