What are the two different Jewish groups that developed after the Jewish Diaspora and how are they different?

Answer 1

1) The Sadducees. These were men of politics and secular life, continuing in the ways of the Hellenising Jews. They had abandoned various parts of Judaism; and they claimed no earlier source or tradition for their attitudes. They harassed the Torah-sages; and, like the tiny breakaway group called the Essenes, dwindled away after the time of the Second Destruction, like the earlier Jewish idolaters after the First Destruction.

Note that there is a common conception that the Sadducees, like the much later (and now largely defunct) Karaites, made a deliberate decision to reject the Oral Law and reinterpret the Scriptures.
However, a careful perusal of the Talmud reveals that the Sadducees were actually opportunists who had nothing much at all to do with religion in any fashion. They were lax in Judaism; they were men of politics who had little interest in Torah-matters.

At that time the Jewish courts still had the ability to enforce the Torah laws, and almost all Jews were Torah-observant; so, in order to avoid total rejection by the surrounding community, the Sadducees outwardly maintained a facade of keeping the major Torah precepts (such as the Sabbath), while ignoring the Oral Torah and customs and flouting the words of the Sages.
They went lost not long after.
The group that did (on rare occasions) debate against the Torah-Sages concerning subjects of religious observance, were a tiny sect called the Baitusim (Boethusians), who quickly died out.

2) The Essenes were a small sect in Judea who eventually disappeared from the Jewish community. They styled themselves "observant; pious ones." The normative, majority Jewish community viewed them as breakaways from the common stream of Jewish tradition, because of their beliefs and practices.

Their beliefs included an excessive amount of dabbling with the names of angels, messianic fervor, gnosticism and eschatological speculation; and their practices were more like Christian monasticism than the generally accepted Jewish way of living.

The practices of the Essenes included vegetarianism, dwelling in isolated groups, communal ownership, monastic asceticism and avoidance of money, commerce or private property; and (among some of them) celibacy. Also, they had some forms of non-traditional observances (such as round phylacteries [tefillin]). Some researchers identify the Essenes as a form of early Christianity, taking also into account the fact that early Christianity was far from uniform and was, for a time, thought of by some as a kind of modified Judaism.

3) The Karaites are a small group (about 0.3% of world Jewry), begun about 1250 years ago, that reject the Oral Torah and original (Rabbinic) traditions of Judaism. They are thus forced into the conundrum of needing, in any case, to create their own interpretations of what the brief Bible-verses mean. Example: not working on the Sabbath day (Exodus ch.20). Without a tradition, this could mean almost anything. Rabbinical Judaism understands this command in terms of the ancient tradition recorded in the Talmud (Shabbat 73b), which spells out exactly which activities are called "work." The Karaites have to come up with their own ideas concerning this and thousands of other examples. In doing so, they adopted some rules (such as those of Sabbath) from the Samaritans (no lights or heated food), abandoned other observances (they have no Hanukkah, mezuzah, or tefillin), or altered them (such as making Purim into a fast-day). Some things were made more permissive (allowing meat with milk), while others became completely forbidden (no doctors or medicine).

While one thousand years ago Karaism became relatively widespread, it gradually diminished, as some Karaites converted to Islam and others assimilated. Many Karaites returned to traditional Judaism, under the influence of such Rabbinical luminaries as Saadia Gaon and Maimonides.

Answer 2

After the Diaspora, there were an incredible variety of Jewish groups the formed. Probably the two you are looking for are Ashkenazim and Sephardim, which represent the Jews of Christendom and the Jews of the Islamic World respectively. Some differences in dress and minor traditions developed between the two general communities.