All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The difference between the Children of Israel and the Jews is that the children of Israel are a race and not a religion whereas Judaism is a religion and not a race. The Children of Israel are attributed to Israel who is also Ya’coob (Jacob) and Allaah called him as Ya’coob in many verses and he mentioned him as Israel once in Chapter Aal-Imraan; Allaah Says (what means): {All food was lawful to the Children of Israel except what Israel [i.e. Jacob] had made unlawful to himself before the Torah was revealed.”}[Quran 3:93]
With regard to the second question, then the contemporary Jews and Christians are considered People of the Book as we have already clarified this in Fatwa 82067.
Also, for more benefit, please refer to Fatwa 91581.
Allaah Knows best.