Literally, Aharon stood between the dead and the living. The Baba Sali suggests that this pasuk refers to Aharon’s advocacy on behalf of the living, rather than his standing between them and preventing the Malach Ha’Maves, Angel of Death, from completing his mission. Aharon prayed to Hashem concerning the distinction between the living and those who have passed from this world. The living have the opportunity to serve Hashem, study His Torah and observe His mitzvos. The dead no longer have this opportunity: Lo ha’meisim yehallelu Kah, “Neither the dead can praise G-d” (Tehillim 116:7).
This is how Aharon was able to swing the balance of judgment in favor of those who were still alive. He asked Hashem what was to be gained by their deaths. They would neither be able to serve Hashem any longer, nor would they be able to repent their sins. By keeping them alive, the chance for teshuvah, repentance, increased exponentially.
While this is a powerful argument on behalf of the living, one would thus assume that it should be applied equally to everyone – regardless of his sin. We see that Moshe Rabbeinu obviously did not feel this way as he asked Hashem, Al teifan el minchasam, “Do not turn to their gift-offering” (ibid 16:15). Did the two leaders have divergent perspectives on outreach to sinners: Aharon prays for life, whereas Moshe asks Hashem to ignore their sacrifice?
The difference is in the identity of the sinner and the egregious nature of the sin. Moshe spoke concerning the leaders of the mutiny, the rabble-rousers who sought to undermine his leadership and impugn Hashem’s authority. They had gone too far. Their rebelliousness was of such a nature that teshuvah was but a dream. Aharon, however, was addressing the hamon am, common folk, who were misguided and often followed those who made the most noise. They had a chance to return, because they were not sinners; they were just disillusioned people who had been misled by a demagogue seeking honor for himself.
