Yes, they can make a significant difference in helping improve your performance and helping decrease your chances of injury, from wearing either poorly made shoes or shoes that simply aren't the proper style for your body, gait, and type of running.
If you're going to run or participate in any high impact sport or sport that will put a lot of stress on your body, then it's worth it to invest in the proper attire, footwear, and/or accessories. For any high impact sport, the right shoes for you and for the activity you're participating in, can mean the difference in enjoying the sport or sitting on the sidelines with an injury.
Different people have their fave brands of running shoes, but ultimately only you can decide what feels best for you. I suggest going to your local running apparel store and having the pros there evaluate your gait (walking and running) to help determine what type of shoe is best for you (neutral, stability, etc) and from there you can try various brands to see which feels the best on your feet. I've had success with stability shoes from Brooks, Hoka, and just bought a pair of Topo but that doesn't mean any of those brands will be your preference.