The Aaronic Priesthood still exists in many quarters of the Jewish world, but it is a separate institution from Rabbis. Descendants of Aaron (called "Kohanim" in Hebrew) were the priestly caste in the times of the Temple, as you noted. When the Temple was destroyed their position and responsibilities were diminished considerably.

However, there are still some remnants of it. First, a disclaimer--some of the more liberal denominations of Judaism have eliminated the institution within their denomination, or substantially reduced the privileges they get, so this doesn't necessarily hold for every part of the Jewish world. But what I'm writing is true of pre-modern Judaism, and some quarters of modern Judaism. Certain rituals/honors are reserved for a Kohen, if one is present, and others for a Levite (which is the larger quasi-priest group of which Kohanim are a subset). They perform the priestly blessing, either on holidays only (in the European Jewish world outside Israel) or daily (in Israel and among non-European Jews). They perform certain rituals, notably the redeeming of the first born. They also have some additional marriage restrictions. And notably, they are forbidden from entering cemeteries or having contact with the dead.

But once the Temple was destroyed, the bulk of their source of authority/power was eliminated. The Rabbis (whose exact role in the Jewish world was kind of unclear) eventually were able to claim the religious leadership role of the Kohanim, and without a political power system the Rabbis were sort of all that was left. Despite that, there are almost no formal liturgical privileges for Rabbis. It's a title, and communities now tend to hire at least one, but there are virtually no rituals that actually require a Rabbi. Unlike Kohanim, who still have special ritual roles.

Of course, there are Rabbis who are also Kohanim, but there's no particular reason for them to be connected. It can actually be more difficult to combine the roles, since if a Rabbi can't enter a cemetery it makes it impractical for Kohen-Rabbis to officiate at funerals.

I have no idea if its scholarship is still current, but Stuart A Cohen's "The Three Crowns" talks about formalization of Rabbinic power, specifically over other power structures that previously existed in Judaism (namely Kohanim, who still exist but are not a power structure, and political leaders, who did exist after the Temple's destruction but are not a power bloc per se).

There is a difference between the Israelite priests claiming descent from Aaron, and rabbis. The priesthood (or kohanim) is an older office than the rabbi.

In ancient Judaism, the priests were responsible for performing religious ritual and sacrifice at the Temple in Jerusalem. According to the Hebrew Bible, the Temple housed the Ark of the Covenant, which was built by the Israelites under the leadership of Moses. Traditionally, the office of high priest is seen as beginning with the inauguration of Moses’ brother Aaron. The Hebrew Bible also states that the high priests served in religious duties pertaining to the tabernacle of Moses, and that the duties continued after the tabernacle and the Ark were moved into the Temple of King Solomon. The priesthood was hereditary, each high priest claimed to be a descendent of Aaron. According to the Torah, when Moses, and later Joshua were instructed to lead the Israelites into the land of Canaan, each of the tribes of Israel were given an allotted part of the land of Canaan, except for the tribe of Levi, rather than being given a portion of the land, the Levites were instructed to perform the religious duties of Israel. The word “Levitical” is often used in a manner of describing the priestly duties of the Hebrews for this reason, (Leviticus, which deals almost entirely with priestly law, literally means “relating to Levi”). Of course, there were more needed to maintain the Temple than just the high priest, others were given the role of cleaning and maintaining the Temple, just like the high priest, however, those who assisted in maintaining the Temple had to be Levites.

After the First Temple was burned by the Babylonians following Nebuchadnezzar II’s capture of Jerusalem in 587 BC, the priesthood was suspended. The Temple, and with it, the tabernacle that was supposed to be a resting place for God, didn’t exist anymore. There was no use for a high priest to perform the religious rituals, or for the Levites to maintain the Temple, as there was no Temple to maintain. There may have still been a few people worshipping at the site of the old Temple, but for the most part, the priesthood was useless during the Babylonian captivity. Following the Persian occupation of Babylon by Cyrus the Great, however, the Jews were brought back to Israel (now Persian Judea or Yehud) and started to rebuild the Temple in 538 BC. The office of the high priest came back with the rebuilding of the Temple in Jerusalem. Again, the priesthood in the Second Temple period was hereditary, the high priests were still Levites claiming descent from Aaron. During the Jewish-Roman Wars in the late first century AD, the Second Temple was burned by the Romans in AD 70, to this day, the Temple has not been rebuilt.

Though there has not been a Jewish high priest since the burning of the Second Temple, some more orthodox sects of Judaism still recognize the authority of priests to give blessings to the Jews. Synagogues that welcome priests today must rely on the testimony of the priest that they are descendants of Aaron. There is no sure way to prove today whether the priests claiming to be Levites are being truthful, as DNA evidence has not grown advanced enough. Aside from the blessings performed, the priests today anticipate the building of a Third Temple in Jerusalem, and practice for the reintroduction of the office of high priest.

Though there are still some synagogues which recognize Jewish priests, the priests are not the same as rabbis. A rabbi (literally meaning “teacher”) is somebody who teaches and interprets Jewish law as presented in the Torah (The Talmud, for example, is a collection of writings by various rabbis debating and interpreting the Torah). Rabbis spend much of their lives intensely studying the Torah and teaching it to students. Unlike the priesthood, rabbis are not part of a hereditary office, any Jewish male (some more progressive branches of Judaism may appoint women) may become a rabbi, it is not limited strictly to Levites or descendants of Aaron. Rabbis have existed since the late Second Temple Period, and existed alongside the Levitical priests. The Pharisees were often recognized as rabbis, and Jesus is also referred to as a rabbi several times throughout the New Testament. In modern Judaism, rabbis are usually seen as more important than priests as they are more relevant to the current day. Since there is no Temple in Jerusalem for priests to perform rituals in, the main source of learning and worship in Judaism today comes from rabbis. The high priests were seen as appointed by God to perform religious duties, as such, the rights of the high priests were passed down from each generation. Rabbinic duties are not necessarily seen as being appointed by God, but the rabbis would usually be the most competent person to perform such duties, as they would know Jewish law better than the layperson.

The traditional Jewish priesthood did end with the burning of the Second Temple in Jerusalem by the Romans, though the priesthood is anticipated by Jews to come back with the reconstruction of a Third Temple in Jerusalem, Jewish priests as of today have very little religious importance. Many religious rituals in modern Judaism are today performed by rabbis, but they are unrelated to the Jewish priesthood.

I hope my answer helps!

Sources

Allan, Nigel. “The Identity Of The Jerusalem Priesthood During The Exile.” The Heythrop Journal, vol. 23, no. 3, 1982, pp. 259–269., doi:10.1111/j.1468-2265.1982.tb00641.x.

“Becoming a Rabbi.” Becoming a Rabbi | American Jewish University, www.aju.edu/ziegler-school-rabbinic-studies/becoming-rabbi.

L, M J. “What Are Kohanim, or Jewish 'Priests'?” My Jewish Learning, www.myjewishlearning.com/article/kohanim-jewish-priests/.

Stökl, Jonathan. “Priests and Levites in the First Century C.E.” Priests and Levites in the First Century C.E., www.bibleodyssey.org/en/passages/related-articles/priests-and-levites-in-the-first-century-ce.

The Editors of Encyclopaedia Britannica. “Temple of Jerusalem.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 3 Apr. 2014, www.britannica.com/topic/Temple-of-Jerusalem.