How are ancient judaism and modern judaism different?

There is a huge difference between ancient Judaism and modern Judaism. Ancient Judaism was centered around Temple sacrifices, and the idea that God was physically closer to the Jewish people in Jerusalem. Today, Judaism is centered around prayer and study, and is not tied to specific location, though the majority of Jews support Israel and many consider themselves Zionists.

There is no difference between ancient Judaism and modern Judaism. The laws are the same interpreted quite differently today though they are based on ancient texts. The customs vary from region to region. If you live outside of the land of Israel the laws regarding Israel don't apply to you. If you do live in Israel today the laws regarding the temple are not practiced today because there is no temple. But the laws that were practiced in ancient times when there was no temple, are the same as today.
Specifically,

Orthodox Judaism is today's form of Judaism that most closely resembles ancient Judaism [actually that is not true in the least. "Orthodox" Judaism was a response to Reform Judaism], and its religious law, halacha, is essentially the same. Conservative Judaism believes that halacha is more flexible than was practiced in ancient times. Reform Judaism is more liberal still.

In ancient times there were alternate forms of Judaism just as there are Reform movements today. One alternate Jewish group from ancient times is Christianity. But American Reform Judaism does not seek to break from Orthodoxy in the way that the early Christians did. German Reform Judaism in some cases did seem to desire that alternate religion path, but that direction was for the most part destroyed during the Holocaust and rejected post-Holocaust.