Yes, you can wear any old t-shirt and shorts during athletic activity, but with the array of sport-specific apparel available today, why should you? Sport specific clothes are designed to enhance your performance and comfort for a particular sport and can make your workouts more efficient and more enjoyable. Here are two examples:
Biking: Close-fitting, padded biking shorts are designed to minimize wind resistance and to avoid extra fabric catching on any part of the bike. And the padding will make the long rides more comfortable! But most importantly, don't get on a bike without a properly fitted helmet.
Running: You hear a lot about running shoes, but don't underestimate the value of sport specific socks for running. Quality running socks are moisture-wicking to keep your feet drier and prevent blisters. They also have padding in high-stress areas, which can reduce your risk of developing painful calluses and reduce the overall stress on your feet.
Be sure to choose the right sports apparel for specific weather conditions. For example, when biking outdoors, you're generating less heat and more wind resistance than when walking or running. If you choose a jacket that's too breathable, all the wind will pass right through it and you may feel too cold. If you are running outdoors in the winter, a good rule of thumb is to dress as if the temperature is about 15 degrees warmer than the thermometer says, and you may want a jacket that is more breathable than what you would wear for biking, so you don't get overheated. You should feel a little chilly when you first step outside, and after a mile or so feel fairly comfortable.