I think the most central difference between the Jews and Christians is that the Jews follow the passage in Deuteronomy where it says to accept the teachings of the sages of Israel and their own parents to heart. God made sure the Jews would listen to their elders by making sure to tell them to do that over and over, as even a cursory look at Proverbs will confirm. The most specific passage is in Deuteronomy 17:8-13.
If a matter eludes you in judgment, between blood and blood, between judgment and judgment, or between lesion and lesion, words of dispute in your cities, then you shall rise and go up to the place the Lord, your God, chooses. 9. And you shall come to the Levitic kohanim and to the judge who will be in those days, and you shall inquire, and they will tell you the words of judgment. 10. And you shall do according to the word they tell you, from the place the Lord will choose, and you shall observe to do according to all they instruct you. 11. According to the law they instruct you and according to the judgment they say to you, you shall do; you shall not divert from the word they tell you, either right or left. 12. And the man who acts intentionally, not obeying the kohen who stands there to serve the Lord, your God, or to the judge that man shall die, and you shall abolish evil from Israel. 13. And all the people shall listen and fear, and they shall no longer act wantonly.
Jews take the Torah serious, whereas the Christians take the Torah only as "background information."
The passages in the NT where the rabbis are insulted, especially those passages where Jesus and Paul mention 'man-made traditions' throw Deuteronomy 17:8-13 into the trashcan, not just the traditions of the rabbis.