When most people think of workout shoes, they tend to think of running shoes, like the ones on the left. The problem is, running shoes make horrible weightlifting shoes. In fact, they may not be the best shoe for running either- I will get to that point, later.
Weight lifting requires good stability from the ground up. The thick, cushy heel of most running shoes provides horrible stability. Also, the cushy sole of a running shoe absorbs ground-reaction forces, therefore affecting your performances. A better choice is a harder soled shoe, like the weightlifter pictured to the right is wearing (shoes on the left). These are Olympic Weightlifting shoes and have hard (rubber and/or wood heel) soles. Now, some HTS clients perform Olympic-style weightlifting lifts, many others perform squats, single-leg exercises and kettlebell lifts. A harder soled shoe is preferred for all of these exercises to more effectively 'connect' to the ground.
Running is a different movement and requires the right shoe. Many HTS clients enjoy distance running, others utilize interval sprints for fat-burning and metabolic conditioning. Traditional running shoes can work, but many times the shoes can mask or worse, can exacerbate a leg/hip alignment or muscle balance problem. For example, many females have a condition called valgus genu, where the knee turns in with running or jumping. This condition can place a lot of stress on the ligaments and meniscus of the knee (one of the factors that leads to the much higher risk of ACL tear in females vs. males). Cushy soled shoes (in addition to poor running and jumping mechanics) can exacerbate the valgus genu and place even more stress on the knee.
An alternative to the standard running shoe, and gradually building more interest, is the shoe-less shoe. Pictured to the right is Vibram's Five Finger running shoe. A bare-minimum shoe (thin sole, no support) to help you run with the proper mechanics. They are effective in changing the activation patterns of muscles in your legs. However, you need to learn proper mechanics before you start running. They will not autimatically fix a lot of the mucular skeletal problems most Americans have and you will be in pain if you don't know what you are doing.
The traditional running shoe was developed ,obviously, for running, but for a lot of training it is not the best choice. The thick, cushy sole creates an artificial environment for the foot and robs the body the of the important tactile perceptive input. Additionally, it adds more instability to the foot. It definately is a poor choice for weight lifting and may be an inferior choice for running, especially when proper running mechanics have been restored. You can learn more about running mechanics is previous posts: here, here, here, and here.