I will start this by saying that Korean history in general is a massive blind spot for me so I cannot say much on that, but in general it is fair to say that the actions of the early Mongol conquests were similar in their intensity and brutality across the massive area they impacted. One could argue that the invasion of the Khwarazmian empire was unusually brutal, and many of the canonical examples of Mongol massacres comes from their destruction of the central Asian cities of Bukhara, Merv, Samarkand and Gurganj where later historians recorded mass executions of nearly the entire male population of the city (valued artisans excluded). However, this should not be taken to mean that the Mongols practiced a light touch in other regions--the sack of modern Beijing was a particularly noteworthy example of mass slaughter and enslavement.

I should digress to define what it means when I say "China" at that time. When the Mongols sacked modern Beijing in 1215 it was not part of "China" per se, rather it was under the control of the Jin Dynasty, which was a "conquest dynasty" ruled by the Jurchen, ancestors of the Manchu who conquered northern China from the Song about a century earlier. The actual identity of the Jin and what is and is not "Chinese" is too complex to go over here, needless to say at this time China was divided mostly between the conquest Jin dynasty and the rump Song dynasty. This is important because the conquest of the Song was not until decades after the conquest of the Jin, and was done by a Mongol empire that decades of seesawing warfare and the experience of ruling north China had changed radically in its aims, organization, and temperament of its rulers.

The conquest of the Song was by no means a picnic and there were certainly instances of massacre and brutality, but by that point Kublai Khan had fully assimilated the titulature and style of Chinese rule and did not present himself as a foreign invader but rather as a possessor of the Mandate of Heaven and thus the head of a legitimate dynasty. As such, he was willing to use co-option and inducements to surrender more extensively than his grandfather Temujin, and his very success was in no small due to sailors and merchants on the Yangtze being so fed up by the Song court that they decided their fortunes were better served under a new dynasty. The Mongol army that actually reached the South China Sea was in many ways more like its opponent than the Mongol army that had conquered the North.

So I think you can argue that the conquest of southern China was done with more diplomacy and less brutality than, say, the conquest of central Asia, although that does not mean "none", and I would personally not wish to experience either. Along with that is the fact that conquest did not mean the end of the Chinese experience with Mongols, and the century of the Yuan would prove those who hoped that the new dynasty would bring better rule sadly mistaken. Even before the final conquests most Song commentators believed that the condition of those who lived north of the Yangtze was one of horrific tyranny, and the steady flow of refugees south suggests this was not based on nothing. I wrote a bit about the style of central Asian rule over Chinese territory across history here and while "across history" is a very dangerous phrase I do think there are certain tendencies that find echoes in many different periods.

Unfortunately the story and my post is not quite over yet, as this is all somewhat separate to the question of twentieth and twenty first century perceptions, regarding which there are two factors I want to briefly touch upon: the general pendulum swing of historiography, and modern nationalism. As to the former, history is a discipline that rewards a contrarian streak, arguably the job of a historian is to examine our understanding of the past, find where it is defective or incomplete, and correct it. And so there has been a tendency to examine the popular perception of the Mongols as being the epitome of bloodthirsty savages and push back on that, focusing on aspects of Mongol culture we might find admirable (such as its relative egalitarianism and open mindedness), the defects in the historical record (prejudices religious and racial) and the enormity of the Mongol unification of the entirety of central Eurasia and beyond, a stunning achievement that with only minor exaggeration can be said to have given birth to the world as we know it today. All of these are true, important, and neccesary to understanding history, but they do leave themselves open to counter pushback, regarding the brutality of the conquests, the vibrancy of the world preceding them, and the style of Mongol rule. My personal sense as an outsider with a deep interest in Chinese and central Asian history is that this latter pushback has been the more dynamic force in historiography for perhaps ten years, ten years from now though, who can say?

Nationalism is another aspect, it is impossible to talk about Genghis Khan without mentioning the efforts of the Republic of Mongolia to burnish his reputation. This is an entirely normal thing for a modern nation state to do: Greece, for example, has been in a a diplomatic row for decades over the national identity of someone who traditionally is credited with ending the classical period of Greek history, sacked one of Greece's greatest cities and was the target of one of the greatest authors in Greek history who is himself a hero of the Greek nation. Still, there is no denying the impact modern politics has played on the perception of Mongol history.

To avoid writing even more, I will recommend James Waterson's Defending Heaven as an excellent narrative history of the Mongol conquest of China, although I would not be surprised if there were plenty of works in Korean. Also I had to really torture this answer to avoid it being even longer than it is, so I am happy to expand on or clarify any aspects of it that needs it.