The American Republic vs European Democracy

Below, I attempt to analyze two examples that demonstrate the fundamental difference between a democracy and a constitutional republic.

Separation of powers. The U.S. is a presidential republic, where the executive, legislative and judicial branches are absolutely independent from each other and do not report to each other. By dividing government, this system weakens it and prevents it from growing strong enough to overstep constitutional bounds.
Most European governments are parliamentary systems - i.e. absolute democracies where the parliament has virtually unlimited power. It is restricted neither by the executive (presidents and monarchs are nominal figureheads, while prime ministers represent the parliamentary majority), nor by the courts, which have much less power than the U.S. judicial system. The main principle of despotism is to centralize as much power in one body as possible. Whether this undivided and absolute power is wielded by a monarch or a democratic majority does not matter. There have been attempts to introduce at least some degree of separation of powers but they were inefficient. The legislative branch remains supreme, with all other authorities being subordinate to it. Some countries (such as France) have switched to a compromise between the parliamentary and presidential systems (a semi-presidential republic), but the high degree of power centralization still prevents a limited government from emerging.

The independence of the judiciary. The contrast between the American and European judiciary systems largely stems from, respectively, their common-law and civil-law traditions. So the U.K. and Ireland are exceptions, though their parliamentary systems make the judiciary less independent than in the U.S.
In the U.S., a life tenure for judges has been a long-running tradition that is intended to prevent executive influence on the judicial system. But this is not enough for making judges fully independent. U.S. judges determine law according to precedent and are not restricted to interpreting statutory law passed by legislatures. Anglo-Saxon common law (as opposed to statutory law) is, as it were, a realm independent from legislative assemblies, "a law unto itself." Though legislatures still adopt statutory laws, the judicial system's common-law roots nonetheless make it an independent source of power, not a subservient clique of bureaucrats. As if this power were insufficient to restrict the executive and legislative branches, the U.S. political system put yet another obstacle in the way of tyranny - juries. Such is the importance of this institution that the concept of "jury nullification" emerged - the idea that juries have a right to judge not only the facts but also the law - i.e. nullify unjust and tyrannical laws (this power, as any other, can be abused but, if applied properly, it is a check on government power). Moreover, in the U.S. adversarial system, the judicial process is largely driven by the parties themselves, represented by their attorneys (not only by judges), which puts yet another important check on the power of government.
In Continental Europe, judges used to be bureaucrats who could be appointed or dismissed at the whim of a monarch. This is no longer the case but traces of their subordinate and secondary role remain. Unlike in common-law systems, judges cannot "make law" and can only interpret acts passed by legislatures. The jury trial is much less important in civil-law countries and even absent in some of them. Civil-law courts are based on the inquisitorial system, where judges (appointed by the executive or legislative branches) play a dominant role, while attorneys ("the private-sector" element, as it were) are subordinate.

The bicameral legislature. In the U.S. constitutional structure, the House of Representatives and the Senate have equal powers and balance each other. The House is traditionally more "popular" and represents the union, while the Senate is more "aristocratic" and conservative and represents the states. The Senate has historically impeded the progress of democratic tyranny. One of the glorious Senate traditions that thwarted the onslaught of statist legislation has been the filibuster.
In Europe, the principle of bicameralism is not strictly observed. In Portugal and Scandinavia, the legislatures are unicameral. In the rest of Europe, they are mostly bicameral but the upper house usually has far less authority than the lower one and is not a real check on its power. In the U.K., the House of Lords has been gradually emasculated and turned into a rubberstamping shadow of its former glory. The trend toward unicameralism is a manifestation of power centralization and unlimited government.

Federalism. The division of authority between the states and the union is the cornerstone of the U.S. constitutional system. Separation of powers between the three federal branches would perhaps be insufficient to rein in government appetites. Federalism puts another important check on the powers of government. Originally U.S. federalism was so strong that states' rights to secession and nullification were asserted. Unfortunately, these rights have been largely lost as the U.S. government became less limited.
Most European countries are unitary states. Some are federations but much less pronounced ones than the U.S. States generally have less authority and sway than in America. Germany's Bundesrat, which is supposed to represent states' interests on the federal level, has authority only over some issues and cannot affect others. Moreover, votes are not equally allocated for each state and different states' delegates have different voting powers. Recently, however, there has been a certain trend towards "federalization" - Belgium became a federation, while regions received more autonomy in France and Spain, which still remain unitary governments, nonetheless.

The Constitution. Though there had been similar documents in the past (American colonial charters, England's Instrument of Government of 1653, the Swedish Instrument of Government of 1772, the Corsican Constitution of 1755), U.S. state constitutions and the U.S. federal constitution were perhaps the first constitutions in the modern sense. Ironically, they have been the most enduring ones, and the U.S. constitution has not been replaced by another one even once. Nowhere is the respect for constitution so entrenched as in America.
In Europe, constitutionalism has fared much worse. The U.K., though the homeland of modern constitutionalism, has so far failed to pass a written constitution, which may be attributed to the parliament's unwillingness to subject its arbitrary power to any restrictions. Moreover, much of the original "law of the land" and the original "rights of Englishmen" - the cornerstone of the ancient English constitution - have been effectively repealed. France, Sweden and the Netherlands have had constitutions since the 18th century. In most of the rest of Europe, constitutional government was only firmly established around the mid-19th century (though there had been earlier abortive attempts to introduce constitutions).
European constitutions tend to have been frequently amended, replaced and suspended. The basic principle of constitutionalism - the immutability of law - has thus been totally subverted.

The Bill of Rights. The U.S. Bill of Rights is the fundamental document that makes all other aspects of the constitutional system meaningful, the heart that pumps the blood of the body politic. It is individual rights that are the justification of government authority, and whenever government violates them, as opposed to protecting them, its authority is rendered null and void and the body politic is dissolved. A constitution without a bill of rights is a meaningless document.
In Europe, there are no real equivalents of the U.S. Bill of Rights. Most of the provisions of the English Bill of Rights (1689) have been repealed by the parliament. The French constitution, though it pays lip service to the Declaration of the Rights of Man and of the Citizen (1789), is not actually bound by it. Some other European constitutions (e.g. the Spanish, Italian and German ones) have lists of "basic rights" but it is specifically stated that they are subject to restrictions and can be limited by law.
Nor are the basic rights protected by the U.S. Bill of Rights recognized in Europe. As to the First Amendment, the U.K., Denmark, Norway and Iceland still have established churches, while in other European countries the church and state are still closely intertwined. Unlike in the U.S., there is no absolute guarantee of free speech in European countries, which makes such idiotic stuff as "hate speech" and Holocaust denial laws possible. This also entails state ownership of major media. The right to bear arms is not protected in Europe either. Some European constitutions also stipulate positive rights (e.g. to education, healthcare, labor), the absurd concept that effectively overturns negative rights listed alongside them and renders them meaningless.

To sum up, the US, at least as conceived at its founding, is a federal constitutional republic - a government of laws, and not of men. To a certain extent, it is a limited government.
Most European countries are unitary democracies with ineffective constitutions - governments of men, and not of laws. In a certain degree, they are absolute, unlimited governments.