Publicity, ballyhoo, promotion, propaganda are comparable when they mean either a systematic effort to mold public opinion in respect to something or the means or the matter used in such an effort.
Each implies a specialized form of advertising.
Publicity is used especially in reference to the activities of and the information disseminated by a person or persons in the employ of individuals, corporations, organizations, associations, or institutions that seek advertising through more or less indirect means in order to attract attention to themselves, their products, or their objectives or that wish to provide a source of authoritative information on matters concerning themselves that are of interest to the public; thus, the work of a theatrical press agent and of a public relations counsel is publicity; in the first case, for an actor or producer seeking favorable notices in the press; in the second, for a corporation or institution that seeks to control the kind of information regarding itself that is published.
Ballyhoo is indiscriminately applied to any kind of advertising, publicity, or promotion which the speaker or writer regards as noisy, sensational, insincere, misleading, or unduly obtrusive.
Promotion is specifically applied to the systematic efforts of a business organization to gain advance publicity for something new (as a venture, a product, a motion picture, or an issue of bonds) in order to ensure its favorable reception by the public when it is launched.
Propaganda is applied to the concerted or systematic effort of a group that tries to convert others or to hold others to its way of thinking, and to the means employed and the matter circulated. The term has chiefly derogatory, but occasionally underogatory use.
In derogatory use it frequently implies publicity sought through objectionable, usually underhand, methods or for a cause that cannot work in the open, and with the intent to win over the gullible or the unwary.
In nonderogatory use propaganda often implies the ends of convincing a prejudiced or ignorant public and of inducing it to accept something it is disposed to reject. Even in this use the word usually suggests indirect methods.