Both the American and French Revolutions were inspired by the Enlightenment. Specifically, one could generally compare the two saying the French one is the American counterpart. In many ways, the French Revolution was inspired by the Americans. However in detail, revolutionaries of the two were fighting against different monarchies, different politics, and were under different circumstances. The two revolution's similarities come from their cause and the ideas which inspired them.

The American colonists wanted a more direct representation and a louder voice in politics because they were dissatisfied by British taxation which were implemented with little colonial approval. Likewise, the French revolutionaries were unhappy with the Bourbon monarchy because of the large gap between the poor and the wealthy. In this sense, both revolutions revolted for the monarchies' economic management. But clearly, they are different in detail. The colonists were not dirt poor like many of the French were, nor was there a giant chasm between basically two economic groups in the American colonies. The French drew upon ideas of the American revolution, that citizens have a right to speak out against poor societal and economic standards, and applied them to their circumstances. During the Ancien Regime (the period just before the French Revolution), there were three social class: First (monarchy), Second (clergy), and Third (everyone else) estates. The Third Estate consisted of more than 90% of the French population and most of them were indeed dirt poor. In both revolutions, ideas of classical republicanism, natural freedoms (of religion, expression, and assembly), and liberalism were their inspiration. Like many other events in history, they happened because of a Zeitgeist which prevailed during the time. The American Revolution roughly began in the 1760s and the French Revolution began in the late 1780s; both were about the same time.

Famous Enlightenment thinkers such as Jean-Jacques Rousseau, Voltaire, Denis Diderot and John Locke inspired both revolutions' leaders. Voltaire and Rousseau were open critics of the Ancien Regime and the class inequality brought about. Voltaire argued that citizens require freedom of speech, assembly, and religion to separate themselves from absolute monarchs and overthrow the aristocratic elites. Rousseau argued that absolute monarchies were the main cause of suffering among peasants and the common people. His political philosophy of classical republicanism theorized a state that would have the "consent of the governed" (based on his book The Social Contract). For the Americans, many of the Founding Fathers were Enlightenment thinkers. Thomas Jefferson, Benjamin Franklin, James Madison, and Thomas Paine are just a few to begin with. They were political philosophers who were basically derivatives of the European thinkers but also contributed many ideals that are still held by Americans today. The U.S. Constitution, the Bill of Rights, and the Declaration of Independence are complete embodiments of Enlightenment ideals (freedom of religion, speech, assembly, rights of the governed, moral responsibility of the leaders to represent the people, etc.). (I really don't want to sound like a broken record but because the Founding Fathers espoused so many ideas that were fundamentally the same as Voltaire's, Rousseau's, Diderot's, Locke's, and many others', it's nearly impossible to do so.) On the other hand, the French Jacobins were not Enlightenment thinkers because they contributed little, if any, to the humanities and other academic disciplines. They were simply inspired by the liberalism and romanticism of the period. This is where the two revolutions begin to differ in terms of ideology. Most historians label the French revolutionaries as radicals because of the collapse of absolute monarchies and the Reign of Terror that came after. The American Revolution had a very different aftermath compared to the French.

In conclusion, the two revolutions were only similar in their ideological origins. Each of the two were under different societal, economical, and political climates. Their most direct relation would be that the French admired the American revolutionaries, took American ideals, and applied them to the French's situation. Interestingly, it is their similarity which makes them famous. One could look at many other revolutions, such as the Mexican Revolution, yet they were not as successful or meaningful as the American and French incarnations. These two revolutions are the most direct applications of Enlightenment political thought and made the Enlightenment as the modern Western philosophical and cultural canon.

(This question could've easily became one about the influence of the Enlightenment and the origin of Western political thought but I hope I distilled it down to its effects on the Americans and the French without straying too much away by detailing the differences.)

edit: grammar, spelling