Our nails grow since they are meant to be used quite a lot. If they didn't continue to grow continuously during our lifetimes, they would eventually erode away, leaving nothing behind. Which sucks though, because we need them. (If you have a hard time grasping this concept then look at beaver's teeth, claws of lion, tiger, etc)

In our society, we would need to cut our nails because we don't use our hands alot (we use tools, which protect out hands and nails). So to keep our nails short and manegeable, we simply cut them instead of scraping them along harder/rougher surfaces.