While we engaged in two wars which were supposed to be promoting democracy and freedom around the world, and needed the support of the global community to do, politicians and others often used tactics that went against those ideals. While saying we were working to liberate Afghanistan and Iraq from the tyrannies of the Taliban and Saddam Hussein, prisoners we took were tortured for information (which doesn't really work anyway, unless you're Jack Bauer), held for indefinite amounts of time without being charged or having access to lawyers, and many civilians suffered immensely from the invasions (yes they have received benefits too, but some of the casualties and suffering probably could have been avoided). In the eyes of much of the world this directly contradicted our ideals of freedom and democracy, which has turned many countries off from helping us.
Nye also discusses how language which can be used to stir up domestic support may actually diminish international support. While many countries may not approve of the actions of the countries of "axis of evil" many are also not eager to lump "disparate diplomatic situations under a moralistic label" as Nye says. So while they may be interested in supporting the efforts of the US against those regimes, they don't like the language used to describe them and therefore ultimately do not participate in those campaigns with the US. Nye states that the "war on terrorism" is a similar example as it implies a "war of indefinite duration" (did we ever win that "war on drugs"?) and countries were also concerned about its use to justify things like holding prisoners at Guantanamo.
The US clearly wanted to promote their message and win the "hearts and minds" of the people of the world, but unfortunately went about it in a way that turned off many people. Perhaps the American image would have been improved if we attempted more "soft power" approaches before invading Afghanistan and Iraq or if we had incorporated those approaches from the start of the wars.
While Nye indicates the public diplomacy of the cold war was more successful than recent efforts, in some ways I'm not sure why. When the west was fighting communism and the USSR it seems that many of the same tactics were used. There was some extreme propaganda touting the communists as evil (check out the movie "The Red Menace" among some of the other anti-communist films of the era) and an unpopular war to stop its spread. So why is cold war public diplomacy described as being so much more effective than today's efforts? I would theorize that it could be because the enemy being labeled was another country specifically (the USSR) rather than a group of people (or a small section of extremists within a group of people) and that the cold war began right after the west had finished fighting Nazism and fascism in World War II providing a wide base of pre-existing public support. That's only a basic theory, but I'd be interested in what other people think of this dichotomy.