When King Solomon (Prophet Sulaiman) died, the Kingdom of Israel split into 2, the northern Kingdom of 10 tribes retaining the name Israel and the southern Kingdom of the tribes of Judah, Simeon, Benjamin, and some Levites going by the name of Kingdom of Judea. When the Bani Israel living in Judea were exiled to Babylon, it was during this time that the Babylonians starting calling the Judeans, Jews for short, and their religion became known as Judaism. The Bani Israel are made up of 12 tribes and Jews primarily make up 4 of those: Judah, Simeon, Benjamin, and Levi. The Qur’an only makes the distinction between Bani Israel and Jews (Yahud) is when addressing them is the acutal time periods in which they were called by that specific name. So thusly when Allah is addressing them during the time of Moses they are called Bani Israel because at that time that is what they are called and there were all 12 tribes there. At the time of Jesus, Allah addresses them as Jews because in that time period that is what they were called. There is no debate that Jews are Bani Israel. But not all Bani Israel are Jews. Just so you understand that: the 4 tribes of the Jews are part of the 12 tribes of Bani Israel. But not all 12 tribes of the Bani Israel are in the 4 tribes of the Jews.
The Twelve Tribes of Israel’s names are: