Hydrogen peroxide is not as good of a sterilizer as bleach because most bacteria that respire using oxygen, like Staph bacteria, have an enzyme called catalase that breaks the hydrogen peroxide down into oxygen and water. When you put hydrogen peroxide on a cut and see bubbles, this actually means that it's not working because the bacteria are just breaking the hydrogen peroxide down into water and oxygen, which are the bubbles that you see.
With hydrogen peroxide you have to make sure you use an adequate amount to flood the bacteria with a load that's too great for the catalase to handle (i.e. you don't see bubbles anymore). With bleach, only a small amount kills most living things it comes into contact with.