This question can be answered in very broad or very narrow terms.

General principles

In a sense, the most important thing about the American Revolution and Constitution is that they happened at all. The example of a people throwing off one government and replacing it with another, while explaining their reasoning in written documents, is probably America's greatest influence on the world.

For that reason, I'd argue that between the Declaration of Independence and the Constitution, the Declaration has been more influential. Especially since the most admired part of the Declaration, "all men are created equal," is conspicuously absent from the Constitution of 1787.

It's the Declarational ideas of "equality between citizens" and "national self-determination" that contributed to the French Revolution, the independence of the Spanish New World, and the much later wave of 20th century decolonization.

As for general principles in the Constitution that weren't already in the Declaration of Independence, I would highlight these:

• The very idea of a written, self-contained constitution, approved by citizens through a deliberative process.
• The idea that all government offices should be filled by the great body of the people. No institution depends on a special subset of the population, like the nobility or royal family. Election or appointment is the sole basis for legitimacy.
• The idea that political structures should be designed to limit government power. No branch is above the law. There are mechanisms to hold every public official accountable.

Each of these ideas has become pretty much universal today, with only the most diehard authoritarian countries not at least pretending to follow them.

I'd argue that in establishing these principles as global norms, the United States's founding documents have had incredible influence. When it comes to constitutional specifics, however, things get more dicey. I'll next outline the influence (or lack thereof) of more specific parts of our Constitution.

President versus parliament

The US Constitution separates legislative and executive power by creating a President and Congress, who owe their political power to the people, not each other. This is different from a parliamentary system, where the prime minister is chosen from within the legislature.

Worldwide, presidential systems are rarer. They are mostly found in Latin America, where newly independent states really did take direct influence from the United States under the Monroe Doctrine.

Other countries like Canada, India, or Australia, which grew up under Britain's influence, copied Britain's parliamentary system instead. Also, parliamentary systems tended to grow under the shadow of monarchs. For example, after Japan was defeated in WWII, the United States basically wrote the Japanese Constitution under the supervision of General MacArthur. Instead of giving Japan a presidential system like our own, there was a need to respect the immense status of the Emperor Hirohito. A president would have been seen as a rival to the emperor's prestige — unacceptable to the Japanese people — so we gave Japan a parliament.

Historians and political scientists think there's another reason parliamentary systems are more common: on average, presidential systems are more prone to coups, dictatorships, and democratic breakdowns. Juan Linz wrote a famous book, The Perils of Presidentialism, in which he describes how, basically, ever single presidential republic except the United States has suffered from these crises at some point. This makes sense: the executive and legislative are prone to showdowns and gridlock, and both can claim, with equal merit, to be representing the "will of the people." Sooner or later, the military usually gets involved.

Thus, the US Constitution had only a limited impact on structuring the branches of government in foreign constitutions — and, in Latin American countries where it was copied, some might say it was an unfortunate impact.

Federalism

The idea of separate state and national governments, moving in separate orbits, pursuing different tasks, really is an American invention. Before the United States, political theorists used to treat "a state within a state" as an obvious impossibility, a sort of political abomination. The US Constitution, and American political writings like the Federalist Papers, legitimated federalism and developed a political theory behind it.

Fewer than half of countries today have subnational governments, but for those that do, there is definite US influence.

Americans are so used to the idea that each of the 50 states is equal, and all have the same political rights, that they're often surprised to learn that elsewhere, "asymmetric federalism" is common. This is often done to safeguard the rights of geographically concentrated minority groups. For example, the Canadian Constitution gives special privileges to French-speaking Québec not enjoyed by other provinces.

Bill of Rights

The idea of having a Bill of Rights was tremendously important. The specific rights it guarantees, less so.

An unusual amount of our Bill of Rights discusses criminal procedure. Other nations, on average, give fewer specifics as to the criminal justice process. However, foreign bills of rights are more likely to speak about substantive aspects of criminal justice. For example, many include the idea of "proportionality"; the punishment should fit the crime. Our Bill of Rights, which only speaks to procedures, does not say this.

Most nations have no equivalent to our Second Amendment (though it's not unknown).

The biggest difference, however, is the inclusion of "positive rights" guarantees. In other worlds, guarantees of basic human needs such as shelter, education, health care, and the like. Much of this trend owes to the Universal Declaration of Human Rights. Since this was Eleanor Roosevelt's pet project, you can say there's US influence here too, but it's not from our Constitution! (FDR, however, pushed unsuccessfully for a "second bill of rights" late in his presidency to add similar positive rights to our own constitution.)

Things that were not copied

• No foreign democracy uses anything like the US Electoral College.
• No foreign country (for that matter, no US State) has an amendment process as rigorous as ours. The United States is a true outlier in terms of the low frequency of amendments.
• In terms of length, the US Constitution is one of the shortest and vaguest around. Most modern constitutions are far longer, and delve more fine details of institutions. This is often caused by pre-arranged plans for power-sharing between groups. For example, the Indian Constitution reserves certain numbers of seats in its parliament for different castes.

For more information on these topics, I recommend The Endurance of National Constitutions, by Elkins, Ginsburg, and Melton.