Maseches Megillah, Chapter 1, Mishnah 9. This mishnah deals with the distinctions between two different types of Kohen Gadol.
"Ein bein kohen mashuach b'shemen hamishchah lim'rubeh begadim":
There is no difference between the two ways a Kohen Gadol is appointed:
A kohen anointed with the anointing oil - this is the original way of appointing a Kohen Gadol, by anointing him with the shemen hamishchah.
Merubeh begadim - one who becomes Kohen Gadol without the anointing oil, but simply by wearing the additional garments. An ordinary kohen wears four garments and the Kohen Gadol wears eight, and putting on the four extra garments is the other way of becoming a Kohen Gadol.
After King Yoshiyahu hid away the anointing oil, the option of anointing was no longer available, and the appointment of the Kohen Gadol was made through the additional garments alone. Even so, there is no halachic difference between these two types of Kohen Gadol, "ela par haba al kol hamitzvos" - except for the bull brought for all the mitzvos.
A Kohen Gadol who ruled on a halachah in error and acted upon it - similar to a beis din that issued a mistaken ruling - if the matter is one that carries the penalty of kares, he does not bring the sin offering that an ordinary person brings, but rather a special offering: a bull. This offering is brought only by a Kohen Gadol who was anointed with the anointing oil; but a Kohen Gadol of the later generations, who became Kohen Gadol solely by wearing the additional garments, does not bring this offering.
The serving kohen and the kohen who stepped down:
The mishnah discusses a case in which the Kohen Gadol became disqualified, having contracted tumah and being unavailable for the service, and a substitute Kohen Gadol was appointed in his place, after which the original Kohen Gadol returned to his position. The original Kohen Gadol is called "the serving kohen" - the one serving at present; while the one who served temporarily is called "the kohen who stepped down" - a former Kohen Gadol, who is not a Kohen Gadol in the full sense, and yet some of the laws of the Kohen Gadol still apply to him.
These are the words of the mishnah: "Ein bein kohen meshamesh l'kohen she'avar" - there is no halachic difference between the kohen currently serving and the kohen who previously served as a temporary substitute and returned to his former status, except in two matters:
The bull of Yom Kippur - the bull that the Kohen Gadol is obligated to bring from his own funds as part of the Yom Kippur offerings.
The tenth of an eifah - the meal offering known as "minchas chavitin," which is brought every day, morning and evening.
These two offerings are brought only by the Kohen Gadol who is actually serving.
In all other matters, however - the fitness to perform the rest of the Yom Kippur service, the prohibition against marrying a widow, and the prohibition against becoming tamei for his relatives - even the kohen who stepped down, who no longer serves as Kohen Gadol, is bound by these, and the two are equal in them.
In summary: Two distinctions were taught in this mishnah. The first is between a kohen anointed with the anointing oil and a merubeh begadim, and the only difference between them is the bull brought for all the mitzvos. The second is between the serving kohen and the kohen who stepped down, and the only differences between them are the bull of Yom Kippur and the tenth of an eifah, while in the remaining laws - the Yom Kippur service, the prohibition of a widow, and the prohibition of becoming tamei for relatives - the two are equal.