Which Everest Base Camp to Go To?

Different Countries: 
South Base Camp is in Nepal, the North Base Camp is in Tibet. Tibet can be difficult to get a travel visa for. They seem to be sporadic; changing their travel rules so you never know when you will have difficulty getting into the country. Nepal on the other hand does not seem to have that problem

Ease of Access:
You can actually drive to the North (Tibetan) base camp since they have extended the highway so if you don't feel you are in the best of health, you could choose the luxury method.

To get to the South (Nepal) base camp, part of the experience includes a "hair-raising" domestic flight in and out of the small town of Lukla, then a 13 day trek to base camp. If you don't want to take the flight, tack on another 4 days trekking (both ways).



Difference in Altitude:
South (Nepal) base camp is at a higher altitude that the north (Tibet).
  South is at 5,364 metres (17,598 ft)
  North is at 5,150 metres (16,900 ft)