Yesterday, I wrote about unstable surface training with devices such as the BOSU ball. In summary, it has its place, especially in rehab to restore joint stability, but beyond that it will limit your strength training results.
Stable surface training allows for a solid connection between the ground, body, and weight. Martial artists, who train with bare feet, call this 'rooting'. Barefoot training (or wearing hard soled shoes) allows you to have a solid connection to the ground to transfer muscular force. The cushioning of a running shoe acts as an unstable surface and detracts from your ability to transfer force to the floor (and have the floor push back equally; 'for every force there is and an equal and opposite force'- Newton's Third Law of Motion, better known as ground reaction forces).
Why is having a solid connection to the ground so important? Well, if you want to progressively overload your muscles (which you do in order to increase strength), you need a solid foundation. Just as tall buildings are built upon extremely solid bases. Below is a video of me deadlifting 405lbs for three reps without shoes. Do you think I would have the same results standing on a BOSU ball?