Privacy in this case can be replaced with "anonymity", too. The less free speech a country has, the more important anonymity is. But even in super free speech-friendly countries anonymity can be important to hide from the judgement of the masses who might "not get it" because of the status quo culture or whatever.
Sometimes ideas can take years to develop and make their way into a culture. If a person is attacked the moment he wants to plant the seed of an idea, either by the status quo-loving crowd or his or her own government, then progress will happen much more slowly.
Protections for anonymous speech are vital to democratic discourse. Allowing dissenters to shield their identities frees them to express critical minority views . . . Anonymity is a shield from the tyranny of the majority. . . . It thus exemplifies the purpose behind the Bill of Rights and of the First Amendment in particular: to protect unpopular individuals from retaliation . . . at the hand of an intolerant society.
Anonymity is a special case of privacy. Privacy is the ability to control how much you reveal about yourself to others and the circumstances of how you reveal that information. Anonymity is simply the choice to reveal no information about your identity.
To some extent they are the same, and to other extents they are not. That is why they are two different words, with some conceptual overlap. Anonymity is a type of privacy, and privacy (in the US at least) is a concept that appears in the US constitution and in subsequent jursiprudence.
Sometimes ideas can take years to develop and make their way into a culture. If a person is attacked the moment he wants to plant the seed of an idea, either by the status quo-loving crowd or his or her own government, then progress will happen much more slowly.