During both a chemical peel and a microdermabrasion what is happening is a removal of the superficial dead cells of the skin, usually the stratum corneum. Treatments may go deeper than this depending on how aggressive the provider is, however the goal is the same. The methods, however, differ.
With chemical peels we utilize a chemical solution which is applied to the skin to remove the surface tissue of the skin. During a microdermabrasion we use a mechanical method to achieve a similar result. The goal is to rejuvenate and exfoliate the skin.