To accept that Christianity is the natural continuation of Judaism, simply based on the fact that Christianity emerged from Judaism and Jesus was himself Jewish, is misleading. There are many fundamental and substantial differences between the two religions, not least of which is the question of who is Jesus.
Basics of Judaism
Judaism is one of the world's oldest religions, dating back up to 2000 years BC from the time when God first called Abraham to leave his home and follow Him. At that time, God made a covenant with Abraham in which He promised to make Abraham the father of a great and mighty nation and that one day his descendants would inherit the land of Canaan if Abraham followed him.
Through Moses, God gave the law to the people of Israel and the fulfillment of the promises became conditional on obedience to the law. To Jews, the law is called the Torah and comprises the written law (the Old Testament) and the Oral law which was handed down from Moses. There are 613 commandments mentioned in the Torah of which the 10 Commandments are the best known.
Judaism: A monotheistic belief
Christian doctrine defines God as three divine persons. A central belief of Judaism is that there is one God, who is the creator of all that we like and all that we don't. There is no evil force with an ability to create equal to God's. The Christian notion of the Trinity is thus incompatible with the Jewish view that rejects the idea of God having three parts.
The View on Jesus
The central tenet of the Christian religion is the belief that Jesus is the Son of God, part of the Trinity, the savior of souls who is the messiah who came to Earth to absorb the sins of humans and therefore free from sin those who accepted his divinity.
This belief in Jesus as divine revelation made into flesh is a Christian idea that is not acceptable within Judaism. According to the Torah, the messiah will be a human, non-divine person who will restore the physical kingdom of Israel, rebuild the temple in Jerusalem and bring earthly peace. Thus to Jews, Jesus was a human living in Israel around 2000 years ago. As a human, he could not save souls and did not rise from the dead. He did not absorb human sin, as sins can only be atoned for by seeking forgiveness, repenting and correcting one's errors. He did not usher in an era of peace as Jews would claim we can tell by looking around at the world. Jesus did not fulfill the messianic prophecies nor did he embody the personal qualifications of the Messiah.
Thus, no one who is Jewish, either by birth or by conversion, can believe in Jesus as the son of God or as the messiah. For the Jewish people, there is no God but God.
Heaven & Hell
Judaism doesn't have a clear sense of Heaven and Hell, with different places in Hell for different punishments. Rather, the idea is that God uses the afterlife to provide ultimate justice and for the wicked to seek some sort of final redemption.
Judaism does not believe people who are Gentiles will automatically go to Hell or that Jews will automatically go to Heaven on the basis of their belonging to the faith. Rather, individual ethical behavior is what is most important.
Free Will and Jewish Ethics
Original sin is a Christian doctrine that says everyone is born sinful. Judaism, on the other hand, rejects the Christian notion of original sin, the idea that people are bad and cannot remove sin by themselves. Judaism believes that people have both a good and an evil nature within themselves and they have the free will to decide how to act. This freedom of choice provides the basis for Jewish ethics.
The Jewish View on Missionaries
A central belief of Judaism is that all people are God's creation and therefore equal before God. It follows that Judaism does not require that a non-Jew convert to Judaism to obtain salvation as long as he leads an ethical life. In fact, only people who voluntarily desire to join the Jewish people out of true conviction are accepted into the Jewish religion.
Jews have no desire to proselytize to others, no desire to turn them into Jews, and no problems in living side by side with other religions. Only with some Christians who believe they have a mandate to proselytize, do problems emerge.
In the US alone, there are many Christian missionary organizations such as "Jews for Jesus" which have made it their target to evangelize and convert the Jewish people to Christianity, a task that the church has failed at in the last two millennia. They manipulate, misquote, mistranslate, and take out of context portions of the Jewish scriptures to make it appear as if they are talking about Jesus. Some organizations define themselves as "Messianic Jews" or "Hebrew Christians" to hide the fact that what they believe in is actually Christianity. They sometimes prefer to call Jesus by Yeshua, his Hebrew name. Using these terms attempts to lend credibility to the idea that what they are practicing is a form of Judaism, which it is not.