[Which is better: leading the prayer or calling the adhān?]

Answer (and Allah grants success): Some have preferred the imām, and others have preferred the mu’adhdhin. What appears to me—Allah knows best—is that the adhān is superior to leading the prayer, because of the many aḥādīth on its virtue, whereas nothing comparable has been narrated regarding imāmah. Among them is the famous ḥadīth: “The mu’adhdhins will come on the Day of Resurrection with the longest necks,” calling out with the testimony that there is no god but Allah …"; the phrase “longest necks” is a kenning for eminence and honor, so the mu’adhdhins will be among the people of highest rank on that Day.
It has also been said about His saying, “And who is better in speech than one who calls to Allah, does righteous deeds, and says, ‘Indeed, I am of the Muslims’,” [Fuṣṣilat:33] that it was revealed concerning the mu’adhdhins.
If it be said: Imāmah in prayer is among the virtues Allah granted to Prophets and Imāms when present, while the adhān is not so;
It was said to him: Imāmah has great merit, and it is not far-fetched—Allah knows best—that imāmah is a reward hastened in this world for people of virtue, as indicated by His saying: “And We made from among them leaders guiding by Our command when they were patient and [when] they were certain of Our signs.” [As-Sajdah:24]
As for imāmah in ṣalāh, it is narrated from the Prophet (May Allah bless him and his family and grant them peace): “Let the one who is most versed in Allah’s Book lead the people …”
There is no dispute that general leadership and leading the prayer are deserved on the basis of merit, which points to their being a hastened reward.
What clarifies this is that imāmah is an apparent honor, elevation, and exaltation—and there is no reward for imāmah from that aspect; rather, reward is for the works entailed by imāmah.
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1