Angry means "indignant," "wrathful," "inflamed." In idiomatic English, you may be angry about a situation or event, angry at an animal or an inanimate object, and angry with (not at) a person. In precise English, mad has a suggestion of abnormality, of being "disordered in intellect," "insane." A mad person is insane; a mad dog has rabies; mad haste is frenzied; a mad idea is senseless or unwise. The formal word for wrathful is angry. Colloquially (that is, informally), mad is often used to mean "angry," but mad is employed by careful speakers and writers to convey only a sense of disorder or abnormality.
Dictionary of problem words and expressions. Harry Shaw. 1975.