How did the defeat by the Romans affect Jewish History?

Answer 1

After the second Jewish revolt, the emperor Hadrian came down hard on them. They were forbidden to even enter Jerusalem or to practice their religion. Many were enslaved, most were scattered away from their homeland.

they had to leave because of them

Answer 2

1) Part of the Roman defeat was their destroying of the Jewish Temple in Jerusalem. The following changes took place after the Destruction: a) After the Destruction, sacrifices would not be possible. Prayer (which had always existed) would now be the sole offering to God, alongside the Torah-study and performance of mitzvot (Torah-commands), which had also always existed.
b) In addition, there are a number of commands (relating to the sacrifices, the Temple, and the Kohanim) that we would be unable to perform.
c) Also, the Sages took several steps after the Destruction: they recorded the Temple-procedures in Talmud tractates such as Yoma, Midot, Shekalim and Tamid, and they enacted observances to remember the Temple (Talmud, Sukkah 41a) and mourn its destruction (Talmud, Sotah 49a).
d) The Jewish population was dispersing gradually more and more, so the sages soon permitted the writing of the entire Oral Tradition so it wouldn't be forgotten.
e) Certain changes were made in the daily prayers. Instead of praying that God accept the sacrifices, the prayers now asked for the rebuilding of the Temple.

2) The Romans did not force the Jews out of Judea in a single expulsion; but nonetheless, Judea gradually lost most of its Jews, as living there became too harsh.

Those Jews who left Judea went to southern Europe, North Africa, Arabia, the Near East, and (slowly) further afield (especially throughout Europe).

See also:

Jewish history timeline

The diaspora (scattering) was difficult for us:

a) Because certain mitzvot (such as those related to agriculture in the Holy Land) now became suspended for the duration of the exile.

b) Because of the impediment to communication between the various communities. This could lead to the development of rivaling groups, each claiming to be authentic.

(for example: the Karaites.)

c) Because the Jews were now a minority among other nations, who were sometimes quite hostile.

d) Along with the Torah, the Land of Israel (Judea) and the Holy Temple had been central to Judaism. It now was to be seen whether Torah-observance could be maintained on the high level that those ideal factors had been conducive to.

Related topic:
How did the diaspora affect the Jews?
To a certain extent, the diaspora caused the various Jewish communities to take on minor aspects of their host countries.
While all Jews share the same Torah and Talmud, and differences in halakhah (Jewish laws) are relatively small, nonetheless some traits of our regions of residence have rubbed off on us. Examples are the differences in pronunciation and accent (for the Hebrew alphabet), differences in dress, and differences in customs and mannerisms.