Magic, sorcery, witchcraft, witchery, wizardry, alchemy, thaumaturgy are comparable rather than synonymous in their basic senses.
In extended use they are sometimes employed indifferently without regard to the implications of their primary senses and with little distinction from the most inclusive term, magic, but all are capable of being used discriminatingly and with quite distinctive implications.
Magic primarily denotes one of the arts or the body of arts whose practitioners claim supernatural or occult powers (as in calling spirits to their assistance, in performing miracles, in divining the future, and in fixing the destinies of men). In extended use the word denotes a power or influence that produces effects akin to or suggestive of those of magic. Usually it stresses the power to call forth an image, an emotion, or a response from or as if from a void.
Less often it is applied to an art or an artist transcending the natural or explainable.
Sorcery is the form of magic practiced by those who use incantations and charms and cast spells in order to work their usually harmful ends. In extended use it is especially appropriate to suggest an attempt to overpower or enthrall by glamour or artful enchantment.
Witchcraft, witchery, and wizardry in their primary senses suggest powers derived from evil spirits or the use of human beings as the instruments for the accomplishment of Satanic ends, the only difference being that the first two are chiefly applied to the work of women, and the last to that of men. In extended use, however, they often vary in implications.
Witchcraft is sometimes indistinguishable from sorcery, but it more often suggests guile rather than enchantment and wiles rather than spells.
Witchery, on the other hand, occasionally implies either sorcery or guile, but usually stresses a winning grace or an alluring loveliness.
Wizardry suggests a more virile and compelling power to enchant and usually connotes exceptional skill, talent, or creative power in the person who exerts such an influence.
Alchemy is properly classed with magic only because its practitioners claimed mastery of secret forces in nature and the power to work such miracles as the changing of base into precious metals. In discriminative extended use, therefore, it implies transmutation or sometimes transfiguration.
Thaumaturgy basically is applied to performing of miracles and wonders or to the art of wonder-workers (as conjurers or those who profess the power to work miracles). In extended use it is applied to what mystifies and dazzles or is designed to mislead or confuse.