Both the Russian and French revolutions involved so many complex events that attempting to draw parallels will inevitably lead to oversimplification. But broad similarities do emerge.
Socially, both revolutions involved a massive and discontented peasantry that often had very different expectations for the revolution from urban radicals. In France, the Great Fear, or the violent peasant uprising that followed the storming of the Bastille in Paris helped to frighten nobles into accepting reforms. But the peasants also served as a conservative force as Parisian radicals enacted price restrictions on grain and especially measures aimed at weakening the Catholic Church such as the Civil Constitution of the Clergy. In Russia, angry peasants also provided support for both the initial Russian Revolution and the Bolsheviks, but they also reacted negatively to collectivization efforts, and, like French peasants, persisted in venerating the monarch.
Economically, both revolutions were driven by extreme shortages. The French Revolution, while triggered by a fiscal crisis, occurred in the context of a series of poor grain harvests that caused skyrocketing bread prices and widespread misery. The Russian Revolution, on the other hand, was sparked by privations resulting from Russia's involvement in World War I, and particularly by coal and oil shortages, among other things, in Petrograd, the Russian imperial capital.
Politically, both revolutions traveled a similar trajectory in that they began with more moderate phases (though the initial phase of the Russian Revolution did lead to the abdication of Tsar Nicholas). Both entered bloody phases as radical forces (the Bolsheviks in Russia and the Jacobins, backed by urban radicals in France) emerged to gain control. Both revolutions also ended in (or some historians might argue, continued into new phases with) autocratic rule under Napoleon and Lenin/Stalin respectively.