The word Jew is derived from Judah who was the fourth son of Jacob whom God named Israel. From this we see that all Jews are Israelites.
After the time of Solomon, the kingdom of Israel was split into two. The 10 Northern tribes retained the name Israel and the 2 Southern tribes were called Judah.
Israel was taken into captivity by the Assyrians and the northern kingdom ceased to exist. Judah, after 70 years captivity in Babylon, returned to the land. The land became known as Judaea.
The religion, based on the covenant that God had made with Israel at Sinai, became known as Judaism. Those who practice Judaism are known as Jews. The term ‘Jew’ in this way includes all the descendants of Israel.
When Jesus and his apostles preached in Judaea they were preaching to the lost sheep of the house of Israel.
Matthew 15:24 But he answered and said, I am not sent but unto the lost sheep of the house of Israel.
Acts 2:36 Therefore let all the house of Israel know assuredly, that God hath made that same Jesus, whom ye have crucified, both Lord and Christ.
Jesus, king of the Jews is the king of Israel.
Mark 15:26 And the superscription of his accusation was written over, THE KING OF THE JEWS… 31… the chief priests mocking said among themselves with the scribes, He saved others; himself he cannot save. 32 Let Christ the King of Israel descend now from the cross, that we may see and believe. And they that were crucified with him reviled him.
When the Jews established an independent state in 1948 they chose the name Israel to include all the descendants of Jacob. Therefore, For all practical purposes, the terms Jew and Israelite are interchangeable.
I hope you find this helpful.