1. We'd be more vulnerable to asteroids
First of all, if the moon were destroyed violently enough for it to break apart, we'd likely experience a pretty violent moon rock storm down here on Earth. Plus, ever wonder how the moon got all those huge craters? It's a veritable punching bag for big ol' asteroids, stopping some that might otherwise make their way down here. Without a protector, we'd be in the line of fire for more asteroids than we are now, which as any dinosaur will tell you, could be very, very bad.