Why did Moshe fall on his face?
Rashi: He fell on his face out of helplessness - due to the Machlokes. He Davened for Yisrael after the Eigel (Sh'mos 32:1), the Mis'on'nim (11:2) and the Meraglim (14:13). This was the fourth sin, and he was afraid that Hashem would not not accept his prayer on behalf of Yisrael. 1
Ramban (in Pasuk 5, citing the Ibn Ezra) and Rashbam: To pray for guidance - and that is when Hashem told him in Pasuk 5, what to answer Korach. 2
Targum Yonasan: It was out of embarrassment - when he heard that they all warned their wives against being secluded with him, 3 because they suspected him of adultery. 4
Oznayim la'Torah: Because until now - by the sin of the Eigel and of the spies, he always had the backing of the B'nei Levi, but this time, the chief rebel was a Ben Levi, and some of his cronies were Levi'im too. 5
Because until now, as long as it was Hashem's Honor that was at stake, Moshe felt capable of defending Hashem's Honor and by pointing out what Hehas done for them in the past; but now that Korach was attacking his (Moshe's) own honor, he felt helpless, and fell on his face to ask for guidance from Hashem how to proceed. 6
Rashi compares this to the friend of a king who interceded on behalf of the errant prince three times to save him the king's wrath. When the prince sinned the fourth time, he exclaimed 'How many time can I trouble the king?' See Oznayim la'Torah who quotes the Pasuk in Amos, 2:4 "On three ytabsgresions of Yisrael I have looked away, on the fourfth, I will not forgive them!'
See Ramban Ibid. DH 've'Hinei'.
See Torah Temimah, citing Sanhedrin, 110a and Note 6.
See Ba'al ha'Turim.
Oznyim la'Torah: As we see from Moshe's statement - in Pasuk 7 "Rav lachem B'nei Levi!"
Refer to answer #2. See also Oznayim la'Torah, who gives a similar xplanation and elaborates.
Why did only Moshe fall on his face, and not Aharon?
Ramban: Because, due to his deep humility, Aharon conceded that Korach was right, in that he was more fitting to be Kohen Gadol than he (Aharon) was. 1
Refer to 16:4:1:2 & 3.
Ramban: Which explains why, although Aharon followed all Moshe's instructions, he did not utter a word for the duration of the rebellion. See also Oznayim la'Torah in Pasukf 2 DH 'Vayakumu Lifnei Moshe'.
QUESTIONS ON RASHI
Rashi writes that he fell on his face, for this was the fourth sin. He fell also at the Meraglim!
Sifsei Chachamim: At the Meraglim, he fell and prayed. Here, it does not say that he prayed, so Rashi needed to give a reason why he fell.
Rashi writes that he fell on his face, for this was the fourth sin. What is special about the fourth?
Moshav Zekenim citing B'chor Shor: It is due to "Al Sheloshah Pish'ei Yisrael v'Al Arba'ah Lo Ashivenu" (Amos 2:6).
Ner Uziel (p.180-183): The Chet ha'Egel involved all three primary bad Midos ? lust, pride, and envy. Chet ha'Mis'on'nim was due to lust. Chet ha'Meraglim was due to pride. And now they sinned again due to envy.
