As with many physical activities, you can over-exert yourself when mining. Lactic acid builds up in the muscles, sweat clouds the vision, and an accidental pickaxe swing can lodge in the head. A mining potion addresses all of these shortcomings. Lactic acid and sweat forms slowly. The likelihood of accidents is reduced.
What it can't do is improve the quality or amount of ore. Pour it in your mouth or pour it on the deposit: it won't matter. No mining potion is going to help you there.