Why did the Jews make changes in their religion while living in Babylon?

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:

Jewish history timeline

Was Judaism altered after the destruction of the Temple?

Answer 2

The Bible indicates that the Jews had recently adopted monotheistic Judaism, during the seventh-century-BCE reign of King Josiah, although some scholars say that they actually adopted monotheism during the Exile and that the scribes retrospectively wrote the story back into the Book of Kings. The Persians were also monotheistic and were most particularly generous to the Jews. Just how much the Jews appreciated the help and support provided by King Cyrus can be seen in Isaiah 45:1: "Thus saith the LORD to his anointed, to Cyrus, whose right hand I have holden, to subdue nations before him; and I will loose the loins of kings, to open before him the two leaved gates; and the gates shall not be shut;" Second Isaiah demonstrates his appreciation, and that of his fellow-Jews, by saying that God himself appreciated Cyrus, although Cyrus has "not known God (45:4,5)."

It was during this period that the Jews adopted many of the concepts previously known only from Zoroastrianism. After the Return from Exile, the Persians sent Nehemiah ostensibly to help the Jewish priests teach the Jews their own religion.