The North Rim is where we prefer to go now.
However, if it's your first visit, definitely go to the South Rim. It has a lot more to do and a bigger area to cover. Most tourists like the busy aspect of the South Rim - perhaps giving them an 'amusement park' type of experience. This is also where the tourist buses drop off hoards of people for a quick look-see. If you do go to the South Rim try to schedule it in Oct/Nov or April/May for a little less congestion. Even the winter months can be awesome and Trailer Village RV in the park will be open. Snow at the canyon is a beautiful experience.
The North Rim is so refreshingly quiet. It's a place to just relax and take in the beauty. There is one Lodge for fine dining and some cabins and a small campground - no hookups. It has some nice hiking but it's geared for a very quiet time. It also has a shorter season because of the higher elevation. It's nice and cool in the summer. It opens in May and closes in October.