Was Jesus' Teaching only for the Jews, and Paul's for the Gentiles?
No! Both of their teachings were for the whole world.
And whenever there is discrepancy between Jesus' and Paul's messages, Jesus who is God in the flesh, would be my reference point for correctness.
The problem with Protestant doctrine is they could not reconcile what Paul said with what Jesus said resulting in a domino effect of missing so many crucial truths. There is an explanation why some differences between Jesus's and Paul's messages.
God's covenants are not different for Jews and gentiles. Example, whenever a gentile lived among the Jews in old-testament days, he had to be circumcised. And the same today, all Jesus's teachings are for both the Jews and gentiles. It is God's new covenant for the whole world; no two parts about it.
Paul's message to the gentiles seemed different because it was mostly high-level speaking. Example, when Paul said "by faith alone", he meant without additional flesh requirements. This is because the Jews in the area at the time kept poisoning the new Christians with their beliefs - " you still need to be circumcised, you still have to refrain from certain foods, etc.." And Paul found out about it, and strongly stressed the message "...no, it's by faith alone..." But the Protestants took that as the message to the gentiles - "by faith alone, nothing else needed". Which people took as the magical faith; you declare it and move on in life. Ignoring all of Jesus's commands that require daily spiritual obligations, like the Lord's Prayer, the vine and the branches, the seeking of God always.
Jesus' teaching was for the whole world
Jesus said in Matthew 28:20: "Therefore go and make disciples of all nations..., and teaching them to obey everything I have commanded you." That was Jesus after resurrection. There are several internet articles on this subject. This one explained it perfectly. I'll summarize the explanation here, and add my perspective.
There are two distinct periods / roles in Jesus' ministry:
Before His death on the cross, Jesus was God's promised Messiah to the Jews. He had to limit His preaching and miracles to the Jews. But the Protestants ran with these misunderstood Bible references as final answer:
Jesus sent out these twelve after giving them instructions: “Don’t take the road that leads to the Gentiles, and don’t enter any Samaritan town. Instead, go to the lost sheep of the house of Israel.
He replied, “I was sent only to the lost sheep of the house of Israel.”
Jesus was given "all authority in heaven and on earth..." Effectively, He was no longer a Jew!
When they saw Him, they worshiped Him, but some doubted. Then Jesus came to them and said, “All authority in heaven and on earth has been given to Me. Therefore go and make disciples of all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Spirit, and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age.”
Notes: This was Jesus after resurrection, telling His Jewish disciples to "go and make disciples of all nations" (not just the Jews), "and teach them to obey everything Jesus had commanded them".
This reference provides a strong undisputable answer to all those who misunderstood the Scripture, thinking Jesus' teaching was for the Jews, and Paul taught the gentiles! With this clarification there is no excuse for protestants to continue in their ways anymore.
Everything God the Son, Jesus Christ, said as documented in the 4 Gospels, applies to the whole world. Everything! No exclusion! From the eating of His flesh and drinking of His blood, to the Lord's Prayer with the "bread" meaning His flesh, to the rest of the scripture in the 4 Gospels, all apply to the whole world, including Christians and non-Christians - "all nations"!
His helper, the human Paul, filled with the Holy Spirit, taught the same Jesus message, but making high-level statements at times as he did not have access to the details in the 4 Gospels, which were written and published after his death. Therefore Paul could not even reference verses from the New Testament.
God's personality is consistent throughout the Bible. Seeking God through reading the scripture clarifies many attributes about Him. We really did not need the explanation on this page to see this clearly. God comes to earth in the flesh, the grandest event in Christianity. And some people think all His parables, teachings, commandments, were only meant for the Jews? Or if not all, some churches pick and choose from His teachings whatever fits their agenda, and the rest "was meant for the Jews!"