How does Post-Exilic Judaism differ from Pre-Exilic Judaism?

Answer 1

Judaism centers around Torah-observance. This has never changed. The Torah contains many hundreds of commands, hundreds of which are applicable today too. When offering sacrifices was possible, they were offered; and when the Temple was destroyed, those particular commands of the Torah were suspended. It's like a man who lost his arms: he doesn't put tefillin on, but his Judaism hasn't changed.
To answer the question:
While the Torah, its commands and the principles and beliefs of Judaism remain unchanged, Judaism has evolved as circumstances have necessitated.
1) After the end of prophecy (some 2350 years ago), the canon of the Tanakh was sealed by a special Sanhedrin (Rabbinical court). It was the same Sanhedrin which placed our prayers in their permanent form (Talmud, Berakhot 33a).
2) Purim and Hanukkah were instituted after the relevant events.
3) Certain fasts were instituted in connection with the Destruction of the Temple.
4) After the Destruction, the laws of sacrifices were suspended.
5) When circumstances made it impossible, the New Moon was no longer proclaimed by testimony; rather, the fixed calendar was instituted (around 360 CE).
6) The Talmud was put in writing (around 500 CE) when it became too hard to be learned by heart.
7) There are seven formal Rabbinical commands. These are:
Saying the blessings over food (and on various occasions)
Washing one's hands before eating bread
Lighting the Hanukkah-menorah
The Eruv
Saying the Hallel prayer on certain occasions
Lighting the Sabbath candles
Reading the Megillat Esther on Purim.
In addition, there are many Rabbinical decrees, mostly from the Men of the Great Assembly (4th century BCE), as well as later enactments.
The purpose of every one of these is to provide a "fence around the Torah," meaning to shore up something that can benefit from strengthening. An example: not handling electric appliances on the Sabbath, even if they are not connected to any electric socket.

See also:

Was Judaism altered after the destruction of the Temple?

Were the Israelites monotheistic?
Connection with Zoroastrianism?

Answer 2

Post-Exilic Judaism adopted many concepts that parallel the beliefs taught in Zoroastrianism, the religion of the Jews' Persian benefactors. This included belief in angels and Satan, although for the most part, Judaism never regarded Satan as evil. It also included belief in heaven as a place of reward. Hell, as a place of punishment, was adopted only to a limited extent. Pre-Exilic Judaism had the concept of sheol, a place that was neither reward nor punishment but of semi-consciousness.

During the early monarchy, Hebrew belief was undoubtedly polytheistic in both Israel and Judah, with ample evidence not only in The Bible, but through Archaeology. Inscriptions discovered at Kuntillet Ajrud and Khirbet el-Qom suggest that the goddess Asherah was God's partner.

During the late monarchy, the use of idols and icons disappeared, as Judaism became 'aniconic'. Judaism was also moving towards monolatry - the belief that there are many true gods, but only one God needs to be worshipped. Nevertheless, the Book of Ezekiel demonstrates that polytheism had survived up to the time of the Babylonian Exile. Numerous cult figurines also attest to a cult of a fertility goddess, probably Asherah. Post-Exilic Judaism was strongly monotheistic, although Proverbs and other Wisdom books of the period can be taken as evidence of the veneration of a spirit/goddess known as Wisdom, probably a successor to Asherah.

Both human and animal sacrifices were offered prior to the Exile, although human sacrifices appears to have been somewhat rare towards the end of the monarchical period. After the Exile, animal sacrifices continued to be important to Judaism, but the former practice of human sacrifice was a matter of some embarrassment.

After the destruction of the Temple in 70 CE, Second-Temple Judaism ceased to exist and two new and very different religious traditions evolved, as rabbinic Judaism and Christianity. Animal sacrifices could no longer be offered, the priesthood was disbanded, and only the Pharisee sect survived as the rabbis of modern Judaism.