One of the lasting legacies of the French Revolution is the absolute need for some level of executive authority. The response to the monarchy before the Revolution was to establish and locate power in the hands of "the people." While the assertion of democracy was wonderful, it soon devolved into a sense of chaos and a lack of control. This led to abuses in the name of "the people" seen in the Reign of Terror. This actually led people to believe that democracy itself was almost as bad as the horrors of the monarchy. In some respects, a similar situation resulted when the United States intervened in Iraq and removed Saddam Hussein from power. While power was wrestled from an autocratic ruler into the hands of "the people," chaos and absolute terror ended up resulting because of the lack of control and executive authority. In the final analysis, the people can have power within some type of structure which allows the best of "the people" to emerge while minimizing the worst of lawlessness and a lack of control.
To me, the valuable lesson taught by the French Revolution is that unbridled democracy and unlimited government are dangerous.
During the French Revolution, the revolutionaries unleashed a hellish system where anyone they disliked was likely to be executed. There was no real system of checks and balances or of personal liberties. There was no protection against excessive government power.
Because they failed to rein in democracy and the power of the government, the French revolutionaries were unable to create a lasting and stable government. This shows us that stable democratic governments need to be limited so they will not go overboard and abuse those that the majority dislike.