If something's irritating, it annoys you. To most people, tuneless whistling and fingernails on a chalkboard are both irritating sounds.
Irritating things bug you. A little kid kicking the back of your seat on an airplane, a driver who follows you too closely on the highway, an early morning work crew right outside your bedroom window — all of these things can be irritating. A more physical way for something to be irritating is if it causes irritation — like a rash or soreness or pain. Your new watch may be irritating to your skin, leaving a red mark on your wrist.