I have been taught that a Jew is from the tribe of Judah. The rest are Israelites.
The members of the tribe of Judah were also included in the term "Israelites." I believe the term "Jew" first starts to appear about the time of the exile to Babylon. Whether descendants of the other tribes living in Judaea or in the diaspora under the Roman Empire saw themselves as Jews or as Israelites belonging to another tribe, I'm not sure. The Samaritans were mainly descended from the northern tribes (allegedly with intermixture with Gentiles settled there) and they are portrayed as hostile to the Jews in the New Testament.
St. Paul insists on his being of the tribe of Benjamin, but I think he may sometimes include himself among "the Jews." St. Luke has him doing so in a speech at Acts 21.39 and 22.3.