Wearing the wrong shoes can lead to injuries and poor performance.
Make sure you go to a running specialty store to get properly fitted for a shoe that suits your running style and type of foot. They can figure out whether you are a neutral, over or under pronator. (Pronation is basically how your feet strike the ground when you are running.)
You may want to consider buying a second pair of running shoes and rotate the pairs, allowing them to decompress and air out between uses. Wearing your running shoes for running only will also make them last longer.
Your shoes will need to be replaced after about 300 to 500 miles of use to avoid injuries from loss of cushioning.