"O douce Volupte, sans qui, des notre enfance, Le vivre et le mourir nous deviendraient egaux." --La Fontaine Enter Cultural Diplomacy
In 1959, the term "cultural diplomacy" officially entered into the U.S. State Department lexicon.
In that year, in remarks  at the University of Maine recorded in Vital Speeches of the Day, Robert H. Thayer, Special Assistant to the Secretary of State for the Coordination of International Educational and Cultural Relations, stated that:
[F]oreign relationships are no longer relationships between government, or heads of state--foreign relationships are the relationship between people of all countries--and relationships between peoples are governed by the way people think and live, and eat, and feel and this represents the culture of a people; and so today we have in the forefront of the implementation of our foreign policy, CULTURAL DIPLOMACY, and to my mind the most important means of bringing complete mutual understanding between peoples, which in turns compels mutual understanding between governments.... The objective of American cultural diplomacy is to create in the peoples of the world a perfect understanding of the life and culture of America. Also in 1959, a State Department 50-page publication (describing foreign USG foreign outreach program) stated its own definition of cultural diplomacy: "the direct and enduring contact between peoples of different nations" aiming to "help create a better climate of international trust and understanding in which official relations can operate." 
Can the Undefinable Be Defined?
Since these two Cold War definitions, countless others have appeared worldwide, all trying to clarify what has become, according to cultural diplomacy specialists C.E. Gienow-Hecht and Mark C. Donfried, "an increasingly perplexing and controversial term." 
Efforts to define cultural diplomacy have included distinguishing it from related forms of communication: propaganda (first highlighted as a term in the 16th century by the Roman Catholic Church)  and public diplomacy, coined in its original Cold War era meaning at the Fletcher School of Law and Diplomacy in the mid-1960s. 
Distinctions have been made between cultural diplomacy and cultural relations. The introduction to distinguished U.S. diplomat/scholar Richard Arndt's study, The First Resort of Kings--a magnificent achievement, charmingly idiosyncratic--states:
"Cultural relations" . (and its synonym--at least in the U.S.--"cultural affairs") means literally the relations between national cultures, those aspects of intellect and education lodged in any society that tends to cross borders and connect with foreign institutions. Cultural relations grow naturally and organically, without government intervention.... [C]ultural diplomacy can only be said to take place when formal diplomats, serving national governments, try to shape and channel this natural flow to advance national interests."  Cultural relations and cultural exchange are also differentiated. According to Nicholas J. Cull, author of widely admired studies about the United States Information Agency, cultural diplomacy "is an actor's attempt to manage the international environment through making its cultural resources and achievement known overseas and/or facilitating cultural transformation abroad." For the online Oxford English Dictionary (OED), it is "the furthering of international relations by cultural exchange."
As for cultural exchange, the OED says it is
a temporary reciprocal exchange of representatives, students, or artists between countries, with the aim of fostering goodwill and mutual understanding." For Cull, it is "an actor's attempt to manage the international environment by sending its citizens overseas and reciprocally accepting citizens from overseas for a period of study and/or acculturation.  What the hell is it, anyway?
Does the lack of a universally accepted definition of cultural diplomacy make it an "awkward or delicate matter"--in other words, a "hot potato"? 
My inclination is to say yes.
The more traditional political diplomacy and economic diplomacy are more straightforward and thus less problematical. The former, according to Thayer, is "the method used to implement our foreign policy." The latter occurred, says Thayer, due to "our rapid economic development," which caused foreign relations to become "involved in trade agreements, customs duties, export and import duties and the like."
The use of military force, an expression of "hard power," can be more acceptable to policymakers and the public than the vague "soft power" with which cultural diplomacy is associated. Does not labeling a program as a war against Something (terror, drugs, poverty) mobilize voters to support it more willingly than calling for activities focused on mutual understanding with other peoples, and especially those who, as George W. Bush put it, "hate our freedoms?" 
"When Defense Secretary Donald Rumsfeld was asked about soft power in 2003, he replied I don't know what it means," soft-power guru Joseph Nye notes in his 2006 essay on the subject. 
In order to justify funding for cultural diplomacy, its advocates as a rule focus on its concrete educational dimension (e.g., the Fulbright program ) rather than its more nebulous cultural ones.
Michael Kaiser, president of the John F. Kennedy Center for the Performing Arts, thinks that teaching arts management to foreigners is a no-nonsense "new approach" to cultural diplomacy, better than taxpayer-provided foreign artistic performances that don't "really" influence citizens by entertaining the elite. 
Even the culturally sensitive Arndt, in a quite recent interview, said cultural diplomacy should be "probably 60 or 70 percent education." 
"Seeing," to quote Thayer's speech again, is "believing"--but with cultural diplomacy you can't see precisely what you're supposed to believe! As the eminent cultural diplomacy scholar Frank Ninkovich writes:
[C]ultural... programs cannot effectively promote narrow national interests (of which the United States has many). That sort of thing must be left to the traditional instruments of foreign policy. The programs themselves, like internationalism more generally, are based at bottom on an act of faith."  The musician Dave Brubeck and his wife Iola, who travelled the world over on jazz tours sponsored by the State Department, provided the essence of cultural diplomacy with this quotation from their musical: "No commodity is quite so strange/As this thing called cultural exchange." 
Culture and Diplomacy
There is a tension between culture, in all its strangeness, and diplomacy, in all its seriousness. To be sure, diplomacy and culture are forms of communication. But they are different kinds of communication. At is most basic, diplomacy consists of negotiations that traditionally were (and still are) held behind closed doors.
On the other hand, culture--highbrow, lowbrow, or seen anthropologically (to quote Thayer again, "the way people think and live, and eat, and feel") is essentially an open conversation or declaration, expressed most strikingly and memorably by artists sharing (and, let's face it, sometimes spitting) their genius at their audiences.
Negotiations, in theory (but, granted, not always in practice), can lead to a signed, formal agreement, whereas the outcome of "mutual understanding," putatively produced by cultural diplomacy, cannot be so clearly measured.  And sometimes, as Ninkovich suggests, the meeting of one culture with another can lead to shock and deracination, not kumbaya people-to-people...