If I had to guess, I would say that most people don’t think about the shoes that they wear to train in – as long as they are following the rules of the gym, they are good. However, weightlifters have been ahead of the curve in this regard for a long time- and figured out that the right kind of shoes make all the difference when it comes to lifting weights. One of the key characteristics of weightlifting shoes is the raised heel- this is what allows for an upright torso when doing squat variations. A raised heel allows more dorsiflexion at the ankle (relative to the ground), lets the knees travel further forward, keeping the hips further forward, requiring less of a bent-over posture to get deep. Look at the two pictures here (These are not of the same lifter, but bear with me for the sake of example):
The one on the left has her heels on the ground, but because of the lack of heel height, is unable to further dorsiflex her ankle, keeping her knees back, forcing her hips back, and leading to a significant lean. The lifter on the right has the raised heel, which allows for the knees to travel further forward, getting the hips forward, and allowing an upright posture (she probably has an easier time getting deep into the squat as well).
Many running shoes have raised heels as well- but the problem is that running shoes are designed to absorb forces- as in, they have give. As soon as you start to load up the bar, the heel that you had with the running shoes can compress, eliminating any positional benefit you might have had. Weightlifting shoes are specifically designed not to have any give. Weightlifting shoes have heels that are made out of wood or a hard plastic to eliminate any give.
The benefit that you get from the more upright posture is less torque on the lower back, (probably) a greater knee extensor activation, and an overall easier time doing exercises that require a very upright posture like front squats or overhead squats. Anyone who has ever dropped their chest doing those exercises knows how difficult it is to recover from that position.
Check out this study: http://www.ncbi.nlm.nih.gov/pubmed/22201687 This some of the research one of our new professors did at his last university. It is worth a read.