We just built a new coop and run 5 months ago. We made a covered run for several reasons. One, to protect them from weather. We wanted them to be able to be outside every day of the year. The roof protects them from snow, rain and the hot summer sun. Two, predators. We have all sorts and feel that our covered run will protect them from just about everything but a bear, which we usually don't have. There was one running around a few months ago but haven't heard of any sightings in a while now so hopefully he moved on. Three, my comfort. I am chronically ill and schlepping out into all sorts of weather to care for the chickens what getting to be very difficult. So we made the coop and covered run as comfortable as possible for me as well as the chickens. The covered run is easier for me to clean and maintain.
In the end, everyone has to do what works best for their budget and situation. Good luck!