Last weekend I went for a hike with my 30 lb kid (plus 5-10 lbs of carrier and gear) on my back and literally ran down a big hill with him on my back (in my VFF KSO shoes!). I couldn't believe how good that felt to be able to do that with no pain!!! I was ecstatic!
Primal running is defined by using a minimalist shoe like Vibram Five Fingers, Feelmax or a racing flat type shoe. Barefoot running (I've only done this for very short distances a couple times) is just as it says.
One BIG note, you need to modify your running style when you run this way. AND, give yourself some transition time... i.e. TAKE IT SLOW. It's like learning how to run all over again. If something hurts, you are doing something wrong!!
Check out these three websites for some great instruction and dialogue on barefoot/primal running:
I also highly recommend getting the book "Born To Run". This has a great story that is centered around minimalist running.