Why did Moshe fall on his face?
Rashi: He fell on his face out of helplessness - due to the Machlokes. He prayed 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. He was afraid that Hashem would not accept his Tefilos on behalf of Yisrael.
Ramban (on Pasuk 5, citing Ibn Ezra) and Rashbam: To pray for guidance - and that is when Hashem told him what to answer Korach (in Pasuk 5). 1
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.
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 Bechor Shor: It is due to "Al Sheloshah Pish'ei Yisrael v'Al Arba'ah Lo Ashivenu" (Amos 2:6).
Ner Uziel (p.180-183): Chet ha'Egel involved all three primary bad Midos