What, if at all, did Jesus teach and do that fundamentally differs from the teachings and actions of other saviour figures, prophets, and gods?

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Forgiveness. If you have to boil down the fundamental difference between Judaism and Christianity, it’s the concept of ultimate and eternal forgiveness.

It pervades all of Christendom. In the Old Testament, if you sinned against the Lord, you’re very lucky if the Lord spares your life. The Lord struck Uzzah dead merely for mishandling the Ark of the Covenant. This sort of thing is why Judaism is almost completely oriented around obeying God’s law.

The Jewish Patriarchs were intended to be the living examples of God’s wishes and plans for humanity. But they were men, and men are flawed. The Old Testament creates the concept of walking with God to demonstrate what sorts of things God wanted to see in men and especially in leaders of men, and illustrates graphically what happens when you fail to walk with God.

The king David was the epitome of the Jewish Patriarch, the one king that God sent down to rule over humanity justly, forever. This does not happen, even though God promised David such, the line of Davidian kings eventually dies out, and that’s where the Old Testament leaves mankind.

The Lord tried twice to make a good king, reluctantly. He didn’t really want to. He said to Saul, the first king God made, that His people were being silly. God delivered them from the Egyptians and gave them their freedom and a land to live on. God was all they needed. This wasn’t enough for the Jews, and they kept clamoring and praying for a king. God gave them one.

The New Testament starts with the accounts of four men who knew Jesus, and the story they tell is the next chapter of God’s plan for humanity. Unlike the Jewish Patriarchs, who were ultimately men, Jesus was actually God’s Son, sent to humanity to act as an example for how men should treat each other and consider their worship. So while the Patriarchs were flawed, Jesus was not. He is the perfect instrument of God’s will on Earth. Christianity is ultimately the effort to be more like Jesus, and receiving forgiveness from God when you inevitably fall short of that ideal.

The missing piece of Jewish theodicy, the necessity of being forgiving, is the constant thread running through the stories of and about Jesus. Jews know all about repentence, the recognition of the need to stop doing evil and to come back to the light, but forgiveness, the ability to recognize that there’s no end to this sort of effort, is much much less present.

Jesus is the sort of king that God really wants as the ruler of mankind. The king that perfectly encourages all people to do well, prosper, help your brother succeed, and overlook faults. Davidian kings could only do well at the sort of things that human kings can do, execute military campaigns, politick, grant favor to those who show worth. Only Jesus could be the king who grants favor to everybody.