More efficient muscle use, less chance of repetitive stress injury. Regular cycling shoes may give you some lateral float, but they lock your foot to the pedal (fore-and-aft wise) in one place, and that's not how we use our feet. When you go up stairs or do leg presses at the gym (efforts not unlike pedaling up a hill), you push with the middle of your foot. Not with the ball of your foot, as you've been told is proper for cycling.
When you run fast, you run on your toes (or off the ball of your foot). When you walk, you land on your heel. Middle-distance runners run off their mid-foot.
Your foot is just a foot, but you use it different ways for different kinds of efforts, and click-in cycling shoes don't let you do that.