Two very different places. Greenville is much more suburban, and Asheville is . . . well, Asheville.
The unemployment picture stinks just about everywhere, so unless you have a significant amount of money to get you by for possibly many months, I agree with the others about not moving without a job lined up (that seems like generally good advice in all conditions, really).
It also depends on what kind of job you are looking for. Greenville certainly has more office type jobs (not sure if Michelin is still the biggest employer in Greenville), while Asheville's strengths are tourism/service industry and health care.
These are just generalizations, of course.
Good luck.