I regularly use the Nike+ IPOD - I needed to calibrate it early on against set distances but now its pretty spot on and consistent. It would be worth testing them both out against a fixed distance and see what you get.
The Garmin uses GPS so is telling you what distance you've covered to a very high degree of accuracy with no need to ever calibrate it. The Nike system tries to count paces and multiplies by whatever you tell it is your average pace length. This can only ever be an estimate and will be more or less accurate depending on whether you calibrate it well, how close to a fixed pace you stick when running, how easy your gait is to interpret as individual paces, ... Short answer is - the Garmin is always right, the Nike might be sometimes.
I have both and it's the Garmin every time. I calibrated my Nike+ kit at a running track and over longer set distances and as the previous posts state; if your pace/gait changes a lot over a run/route then the Nike+ can only approximate the distance covered. My last run the Nike system congratulated me on running my fastest 1/2 marathon, my Garmin correctly confirmed I'd ran an average 10 miler. The Nike+ GPS app for smart phones or Micoach are better but for me the Garmin is still the most accurate and best option!