Huiou Theou, so Zechariah’s was the Aaronic priesthood (which was for ministering), rather than the Levitic priesthood (which was for assistance)?
The levitical tribe as a whole did priestly duties. This included smearing blood on the altar to make the blood holy, draining blood at the foot of the alter, and so forth.
For every sacrifice that was butchered – it was either butchered by the owner themselves, or by a Levite.
The owner did it themselves when they wished purification from sin, etc; They would lay their hands on the animal as a sign that the animal was taking their place with respect to sin/impurtity; The cleansing was to be effected by the life of the person being touched to the altar which then made the blood holy – and symbolically made the blood of the offerer holy as well.
Once butchered, however, my understanding is that a Levite had to do whatever was associated with the altar as a layperson was not allowed to touch it.
For the sanctification of the altar, itself, and what I would call Higher Tasks the sons of AAron were required.
It was a priest of AAron’s line who could take blood from outside the temple and bring it inside to be smeared on the horns of the altar of incense. As far as I know, however, a regular Levi could handle blood to be poured out on the bronze altar outside the tent of meeting/inner temple.
Aaron is of the tribe of Levi; he is the brother of Moses; So an AAronite is a Levi descended from AAron. Moses was essentially high priest, but abdicated much of his office to AAron: Eg: AAron spoke to Pharaoh … etc. and the privelege of high priestly duties then fell to AAron and his sons.
The special privileges associated with AAron over and above a normal Levite, included coordinating the wars and moving of the tribes, burning of special incense compounded for prayers in the temple: eg: Numbers 10:8; 3:4 (wrong incense burned even by AArons sons.caused death, and any incense burned by non AAronites would bring death. etc.)
I am not sure that it is appropriate to distinguish them based on “ministry”, but rather the AAronite priests performed the duties of a higher Honor and (I presume) a better purification.
Notice; it is Moses who washes the AAronites to cleanse them – they do not wash themselves. He was the one closest to God, and his purification evidently held the highest honor.