Both generals were definitely great but reading the comments above, i don't think you guys give Alexander the credits he deserves.
He came from a royal lineage but he also had to fight to win his spurs. Let's not forget that he was in charge of his father's army before he was even 18 and he won his own battles. It's said that he's responsible for beating the Theban Sacred Band, which had earlier destroyed the Spartan phalanx.
He defeated almost without difficulty, the greatest empire the world had seen so far. Actually personally i'd say that defeating the Persian Empire was a much bigger deal than defeating the Chinese or the weakened Roman Empire at Genghis Khan's time.
In at least all the major battles he fought, he was outnumbered by anywhere between 3:1 and 5:1. Still, he destroyed the Persian Empire single handedly, remaining undefeated by the age of 33 having also fought the also large armies of India.
He too had to develop new tactics and strategies to counter and defeat the different types or armies he was facing.. He changed his line formations to defeat scythe chariots and war elephants (which the Greeks had not fought against before) or horse archers, always using the same type of army.. phalanx, auxiliaries and heavy cavalry. He also made the first use of shock cavalry.
He spread not only a language but an entire culture through the East. An entire period of history, the Hellenistic, is basically named because of how deeply Alexander and his spread of Greek (hellenic) culture influenced the world at the time.
As for the fact that Alexander had a government to support him... that government was 3.000 kilometers away and was nothing in comparisson with the colossus of the Persian Empire Alexander was facing. They had more money, way more troops, and the home field advantage, yet Alexander won relatively easily.
Alexander was the measure by which all generals were counted. They say Julius wept when he compared himself with Alexander and Hannibal gave him the first spot when asked by Scipio who he thought the greatest general was. Of course that was before Genghis's time but still, i do believe that Alexander's accomplishments are greater and by that i do not intend to steal anything from Genghis Khan's glory was was himself a beast of a general and leader.