Most importantly the quality of the hay deteriorates (with or without mold) the longer it is in the weather. This means the crude protein drops. Studies have shown a reduction of 30% dry matter in unprotected hay. This means to provide the same nutrient value to the cow when using old weathered hay, as compared to sheltered hay, you should be feeding some grain, protein supplement, etc. to make up for the reduction. Sheltered hay can be stored for years without any major deterioration in quality.
You can have the hay tested for nutrient value at the Extensio office, usually this is free or done for a modest fee. Then you really know what you are doing.
Cover it like Dun says, it's silly not to.