PHYSICAL COMBAT
In hand to hand combat, lets assume that each man was at his peak. Hitler, in 1915, as a soldier in the Imperial German Army, and Genghis Khan in, say, 1185, when he would have been a young man at his physical peak.
Genghis, as a teenager, was already a married man and a respected member of the Mongol community, which meant he was a good horseman and fighter. He, along with some friends, led a raid to rescue his wife from kidnappers, proving that he was both tough and fearless. This moment is said to be the point in his life when Genghis realized he was destined to be a conqueror, making it a good point of reference for proving his prowess as a commander and soldier.
Hitler, on the other hand, fled from Austria to avoid being in an "impure" army, and subsequently failed the Austrian army medical exam when they conscripted him. When WW1 started Hitler petitioned to join the Bavarian Army and permission was granted… by mistake. Hitler did very little fighting once in the war. He was a regimental messenger boy, meaning that he was far behind the front lines, and spent a lot of time with higher officers. This did pave the way for his military learning and the earning of several awards, as close contact with his superiors bestowed upon him a wealth of knowledge and respect. However, despite his standing with those higher than he on the chain of command, Hitler's lack of front line experience likely meant that he was a weakling compared to his tougher, battle hardened comrades.
So, in a hand to hand fight, I would have to say the winner is Genghis Khan.
TIME WAR
Assuming that we transport the Mongol Empire under Genghis Khan and the Nazi Empire under Adolf Hitler into some kind of time void where they can do combat, the undoubted winner would be Hitler. Although technology would definitely be a huge factor, Hitler's victory does not just come from the fact that Germany has guns. The population of the entire world around the time of Genghis Khan was 400 million and his empire only contained a quarter of that, approximately 100 million. The population of Nazi Germany was 90 million. With Hitler controlling nearly as many people in a much smaller area, he has greater ability to mobilize equivalent forces that Genghis would need to rally from across the entirety of Asia.
13th CENTURY DEATH MATCH
Let us assume now, that instead we simply placed Hitler in charge of a medieval state covering the same landmass that he controlled in 1941. This would basically make him a much more powerful version of the Holy Roman Emperor. In this case, I have to say Hitler would be capable of defending himself indefinitely against Mongol invasions, but he would never be able to strike back and conquer Asia. It would be a stalemate. The problem with the Mongol hordes is that they are only really capable in steppe lands and desert regions, and this is because of two factors: wide, flat, open terrain good for maneuvering with horses, and a lack of food/water sources means that it is easier to cripple a nation by striking fast at the critical components of their economy. Western Europe is the opposite of all these things. It is heavily forested, especially Germany, and all of the land is arable and wet. The vastness of Europe's agricultural wealth and the difficulty of their terrain are the historic reasons for why the Mongol conquests stopped in Hungary, and they are the reasons Hitler would find himself in an unassailable position against Genghis Khan.
20th CENTURY DEATH MATCH
Let us now assume the opposite. Placing Genghis Khan as the leader of Central Asia and Northern China in 1941, with Hitler as controller of his historic empire at the time. The first thing you should take note of is the fact that Genghis Khan did not actually rule the Mongol Empire at its height and, in fact, at the time of his death his holdings were confined mostly to the steppes. The most impressive part of the Mongol Empire in 1941 would have to be Manchuria, which holds Genghis Khan's most significant industries and a sizable number of coal mines. Unfortunately, if we consider the political climate of the rest of the world, Manchuria was also envied by Japan, which was much more advanced than the Mongols at the time. Another unfortunate fact is that, in holding Central Asia in 1941, Genghis has cut off Stalin from his valuable hinterlands in Kazakhstan and Siberia, meaning there is no escape from the Nazi hordes during Operation Barbarossa. With nowhere to run, the Russian industries and workers would either be seized or moved to industrial backup sites the arctic, where they would suffer and die. It is very likely that, under these circumstances, the Russians would lose to Hitler. Stalin would be defeated, and Hitler would immediately begin formulating a plan with Tojo for the invasion of Mongolia. In 1945, if Japan had not already invaded, Japan and Germany would make a joint declaration of war against Genghis Khan and in 1946 German Panzers and Japanese soldiers meet each other on the banks of the Onon River.
STATE ANALYSIS
Genghis Khan's Mongol Empire was alike to Hitler's Nazi Empire in only one way: both were dictatorships. In every other way they were different, geared towards different purposes. Hitler wished to eradicate all racial boundaries, merging European culture under Germanic hegemony. Genghis Khan did the opposite, taking up multiple religions to relate best with his diverse subjects. Hitler's economy was geared towards war production, sedentary city living, and industry, whereas Genghis Khan's economy was based upon intercontinental trade, the nomadic lifestyle, and agrarianism. The Mongol Empire was designed, albeit not on purpose, to last a fairly short time, its income based upon taxation, looting, and conquest, and its succession dependent upon the number of sons that the Khan had sired. Hitler was building a nation that he expected to last for thousands of years, and through his purposeful campaigns of resettlement and genocide, he intended on building a foundation for a permanent German state spanning from the Urals to the Atlantic. They can be compared to stars: the Mongol Empire, while larger and brighter, lives only a short time, whereas the Nazi Empire, despite being less impressive in size and magnitude, would live for considerably longer… if it wasn't destroyed by foreign invaders.
While there may be many other ways to measure the strength of these rulers, I believe that the methods I have used are the best that I can provide as a historian. While I do personally prefer Genghis Khan over Hitler (and even voted for Khan in the poll) the clear winner, when comparing the states that they built, would have to be old Adi.