Other personalities in Genesis undergo name changes (Abram/Abraham and Sarai/Sarah), but their new names are used from then on. In Jacob’s case, both his old and new names are used following his name change and in some instances mirror the ups and downs of his spiritual progress.
Within the story of Joseph, Jacob is used 31 times and Israel 20 times. The Word Biblical Commentary points out that “since Jacob is the normal form, it is the exceptional appearance of Israel that needs to be explained.” It goes on to note that while “in prose Jacob always refers to the historical individual, Israel sometimes refers to the people” (see Genesis 46:8; 47:27; 48:20).
Further, “when Israel is used of the individual, it seems to allude to his position as clan head” (see Genesis 43:6, 8, 11; 46:1; 48:2), “whereas Jacob seems to be used where his human weakness is most obvious” (see, for example, Genesis 37:34; 42:4, 36; 47:9).
The commentary explains that the etymology of the two names supports these differences: Jacob means “struggler” or “deceiver,” whereas Israel means “prevailer with God.” Thus “Jacob turns into Israel when his strength revives” (Genesis 45:28; 48:2). Also “in those scenes where Joseph is present, Israel seems to be preferred” (Genesis 37:3, 13; 46:29–30; 48:2, 8, 11, 14, 20–21; 50:2).