Two things: So I'm going to be using the Confucian ideal of Mandate of Heaven, which differs from say the Legalist ideal or the Mohist ideal, simply because that seems to be the most historically popular. And I would suggest asking this in /r/askphilosophy because they would probably be more prepared to give a philosophical analysis of the differences.

The most popular defense of the the Divine Right of Kings comes from Patriarcha by Robert Filmer, subsequently argued against by John Locke, in which Filmer argues that the Divine Right comes from one's lineage from Adam. That is, God appointed Adam directly to rule over the Earth, and Adam appointed one of his sons, then that son appointed on of his sons, such that theoretically one could argue that some king could pronounce his lineage from Adam to himself and claim that since he was appointed by his father, and his father was appointed by his father's father, and so on, we could find that the Divine Right of Adam is passed down to become the Divine Right of that King. This presents some various difficulties which Locke discusses in his First Treatise on Government.

The thing about the Divine Right is that it is meant to disable rebellions and delegitmize those who gain power through rebellion. If God appointed Adam, and that appointment can be traced down to your king, then it reasons that rebellion against that king would be tantamount to rebellion against God. This can be summarized by James the First when he said "The State of Manarchie is the supremest thing upon earth: For Kings are not onely Gods Lieutenants upon earth, and sit upon Gods throne, but even by God himself they are called Gods." Spelling errors aside, this shows exactly the perception of what Divine Right meant.

Now, contrast this with the words of Mengzi who argues that, "When the ruler makes a serious mistake they admonish. If after repeated admonishments he still will not listen, they depose him…. Do not think it strange, Your Majesty. Your Majesty asked his servant a question, and his servant dares not fail to answer it directly." (5B9) and "One who robs co-humanity [ren] you call a robber'; one who robs the right [yi] you call a wrecker'; and one who robs and wrecks you call an `outlaw.’ I have heard that [Wu] punished the outlaw Zhou - I have not heard that he murdered his lord." Finally, "When he was put in charge of sacrifices, the hundred gods delighted in them which is Heaven accepting him. When he was put in charge of affairs, the affairs were in order and the people satisfied with him, which is the people accepting him. Heaven gave it [the state] to him; human beings gave it to him." (5A5) translation source: https://www.iep.utm.edu/mencius/#H4

This is basically how the Mandate of Heaven goes: Are you virtuous enough to win over the people, heaven, and follow the rites? That means you have the mandate of heaven. Are you evil enough that you take away the virtues of the people, and do not follow the rites? You have lost the Mandate and are now nothing more than a common criminal.

So, in summary the Mandate of Heaven is a lot easier to lose than it is for one to lose their Divine Right. One gains the Mandate by being a virtuous ruler and not solely by their lineage, however if one is born into nobility and becomes king it can be assumed that God must have had a hand in that and you have the Divine Right of Kings.

I am going to slightly disagree with /u/xunkuang’s point on Mandate of Heaven. While I entirely agree with the concept that all Chinese dynasties sought to legitimize their rule with emphasis on virtue and winning over the people, virtue/winning over the people is only one aspect of Mandate of Heaven. The other component and perhaps as equally important is a divine "decision", the idea that the fate of dynasties are already pre-determined by the Heaven and cannot be challenged by the human beings.

It should be noted that the concept of Mandate of Heaven was very malleable and was utilized by Chinese rulers in whichever way they saw fit to legitimize their rule. For example, in times such as the North and South Dynasties and the end of Yuan/rise of Ming, a common part of the Mandate would be claiming to "exile barbarians from the Chinese lands". In other times when there has been a protracted era of warfare, a common part of the Mandate would be the recognition that the people throughout China sought peace and unification, and the strongest power in this mess at the time had the duty to achieve that. Many times also, if the previous dynasty was still sufficiently recent and it still held some legitimacy, then a part of the Mandate would just be the ruler's claim of being the legitimate heir of that dynasty.

On the subject of explaining what really is the Mandate of Heaven, Ban Biao, the Han court historian, outlined a great definition, and monarchs of later dynasties would use what he outlined to great effect. While this was some next-level propaganda, it does give us a good understanding of what at least the Han Dynasty believed to be the legitimacy behind their mandate. He argues that “the throne is governed by the Heaven, and cannot be sought by intelligence and power”, “since one’s wealth is pre-determined by fate, the monarchy is also pre-determined”. He gives five reasons to which Han Gaozu was able to gain his mandate: one that he was the descendant of Di (the morally perfect and legendary ruler that supposedly started Chinese history), two that his body was abnormal for the regular man, three that there were omens purporting the Heaven’s support for his rise, four that he was knowledgeable and virtuous to the people, five that he was a good leader who used the right people for the right tasks. So despite virtue being a very essential part of the Mandate of Heaven, so was the understanding that it was largely not of people's control and all that people could do was interpret the omens.

Now let's look at how the Ming Dynasty viewed their Mandate of Heaven, more than 1000 years later. Interesting enough, Zhu Yuan Zhang, the founder of the dynasty, greatly extolled the Yuan Dynasty after his unification, because he believed they once owned the Mandate. He said in his "Announcement to People of Zhong Yuan" that "throughout history's emperors, it was always China ruling over the barbarians, and the barbarians served China as vassals. Never before have we heard of barbarians ruling China and having the Mandate. Since Song's authority fell, Yuan rose to be the master of China, and there was no one that did not disobey them. It was not due to their own strength that they ruled China, but because it was given to them by the Heaven." His belief was that the Mandate was purely a fatalistic matter; the fact that they could have ruled China meant they had the Mandate. This was, however, the result of his realization that the Ming Dynasty also needed to legitimize their own Mandate; like the Yuan, his origins as a beggar was not the most convincing, and he needed to use fatalism as the argument for why his dynasty deserved the Mandate. This was also the reason why he wanted to downgrade Mengzi's importance in Confucian literature.

I noticed that I rambled for a bit so I will cut it here: in short, Mandate of Heaven was a concept that was quite malleable and was used by different Chinese rulers in the way they saw fit in order to legitimize their reign. While the concept of virtue was essential to this legitimacy, so was fatalism and the concept of a pre-determined cycle of power in China.

Wrote this in a rush while also trying to not be caught offtask by my supervisor. I will edit this later to make the whole thing more coherent.