Unfortunately neither of these are actually short histories of the French Revolution. Belloc wrote his a few generations later with the intent of correcting misunderstandings, while Burke wrote his while the Revolution was in process.
Belloc begins with a praise for Rousseau's Social Contract, which he contends that few have read and those who did read had misunderstood it. I read it partially, but roughly have the idea that it presumes that government and social institutions are the result of a agreement among the peoples. My own opinion is that all this was said better by Aristotle with his description of the various forms of government, along with a few other classical authors.
Then there is this thing called Liberty, which is deemed to be a good in itself. Burke rightly notes that the important thing is what people do with Liberty, and wants to wait and see. The classical authors note that the first generation who achieve Liberty will use it for good, while following generations will increasingly see Liberty as a justification for vice. To this Belloc is silent. So which is it? Liberty for good? Or Liberty for evil? Is Fraternity the brotherhood of virtue and vice? Equality the sameness of right and wrong? The classical writers noted that government is called upon to encourage virtue and thwart evil, which is precluded by the non-principles of the French Revolution. Of course nobody really believes in Liberty, Equality and Fraternity if pushed on specific points.
What Belloc does give is a history of the war between the French and the powers around France that resulted from the civil war. This is something I knew nothing about, but follows directly from the practice of the European nations to try to impose a choice of king on a country through war, such as the Spanish Succession and the Austrian Succession. This was not nearly so unequal as the American Revolution, but a disorderly France still had a struggle against the large, organized continental armies. Superior tactics helped the French at crucial points, but for the most part the war was won by throwing masses of peasants into the battles. What strikes me here is that for centuries the Huguenots tried desperately to be able to live their principles out in peace, but were ruthlessly persecuted by their countrymen. Now, in the cause of non-principle, the French finally bother to rise up nearly unanimously and throw off the yoke of a principle that is clearly wrong.
Belloc is a Catholic, yet notes that the Catholic leadership of the times just preceding the Revolution was nearly universally atheist, even though many of the parish priests and church attenders were still Christian. Why is it that Christianity seems doomed to have atheists worming their way to the top of the power structures, then corrupting the institution, robbing the church, and using whatever power they can obtain to destroy others? As noted earlier, the French economy was in a complete mess, but it seems that selling assets of the church and the elites was one of their solutions for gaining money. This I am not so sure about, since I really don't see much in these histories on this subject. I am just starting Burke's work, so this will take some time to finish.