There are more facilities on the south side and the south rim is open all year.
Vegas is a long drive from the GC, if you don't want to camp in the park, south rim again, then either Williams or Flagstaff would be good choice. By the way they are about 60 miles from the village at the south rim. It is a looonnng drive to get from side to side, most decide which side they want to see and then commit. North rim is much less developed and that appeals to many people it is also at a higher elevation than south rim hence why it closes in winter.