Ho boy, a better question might be if there were any aspects of Judaism that were NOT affected by the Exile. As a matter of fact, it is customary to reserve the term "Judaism" (as opposed to Hebrews or Israelites) to the post-exile populations. There are flairs in Jewish studies and history who will be much better able to summarize the changes from the point of view of Jewish tradition; I can mostly speak to the possible role of Persian and Zoroastrian influences.

I have some previously-written up answers that will probably interest you:

• To what extent did Zoroastrianism influence Judaism?

• The Cyrus Cylinder - myth, fact and forgery

The second link is relevant because you are likely to get responses discussing Cyrus, the king of Anshan who ostensibly founded the Achaemenid Empire (in many respects, the empire as we think of it was founded by Darius, the third or fourth Great King, who assassinated at least one of Cyrus' sons and forged a family link to Cyrus). Cyrus is lauded extensively in deutero-Isaiah (a text I go into somewhat in the first link), and he became a legend almost in his own lifetime, to the extent that there is no shortage of literary works detailing his life - unfortunately, they are of virtually nonexistent historical value. He is remembered as an especially just king who let all peoples practice their religion and tradition freely (however, there are no good historical grounds to suppose this, and several to question it). The Achaemenid Empire is viewed in a somewhat different way compared to a few decades ago, but this isn't always reflected in the way it is discussed within religious and other studies. (this is true even within Zoroastrian studies - a cliché goes that books about Iranian history deal with Iranian religion poorly, and vice versa!)

If you have any questions especially about the role of Zoroastrianism or the Persian Empires in the way Judaism developed, therefore, I'm happy to take them.