What is the difference between tachypnea and dyspnea?

Fast vs short. Both terms relate to a pattern of breathing. If you learn latin, you can figure out just about any medical term that exists. Tachy = fast. Pnea = lung or pulmonary or air. Dys = abnormal. Thus 'tachypnea' is rapid breathing. And 'dyspnea' is abnormal breathing (objectively noted by physician) or short of breath (subjectively noted by subject).