In my opinion, it is not really that different than the freedom enjoyed by the citizens of most democratic countries. Of course, Americans have way more freedom than citizens of places like Iran and Saudi Arabia, but not really that much more than those of places like France or Japan.
Citizens of all modern democracies have basic freedoms like ours -- the freedom of speech, the right to vote, things like that. However, you could argue that the US is a bit more aggressive in protecting some of these rights. For example, the right of British people to "speak" is somewhat limited by the fact that it is very easy to sue someone for libel in England. And the right to freedom of religion is under attack to some extent in France where it is illegal to wear religious symbols that are deemed to be too showy.
So Americans don't really have different rights than do people in other democracies, but you can argue that America protects people's individual rights more while other countries are willing to curtail those rights for the (perceived) good of the whole society.
Americans like to think of their country as being based on freedoms and rights, even though you could argue that some countries give their citizens even more freedom than in the US. Some of the specific freedoms in this country are different than you find around the world.
For one, and this is a very American cultural idea, we have the right to own guns without very many restrictions. Most other nations don't understand why this is important to Americans, or why it would need to be in the Constitution. Freedom of religion in the US is extremely well protected, and Koran-burning Pastors notwithstanding, the vast majority of Americans are very religiously tolerant.
We have extensive freedom to travel, without restrictions by the government, requirements for identity or permission papers, and without significant internal border controls. Also, police in the United States have more restrictions on how they can treat citizens, unlike in many other countries.
The United States in known as the "land of the free" but our rights are similar to those of other European countries such as England or France. We do have many more freedoms than some countries. For example, we have more freedoms than some middle eastern countries, especially when it comes to the rights of certain minorities such as women. A lot of this boils down to culture and what is expected of people and it can be very difficult to understand when we are born in a country where "all men are created equal".