They are equivalent. Let me first remind you that a '2D' Hilbert space are described by only 2 independent real parameters. You though it would be 4, but the normalization constraint you mentioned minus 1 out of 4, and a global phase minus another 1, so it's 4-1-1=2.

FYI, the 'global phase' means for the following transformation $$\alpha\to\alpha e^{i\phi}, \beta\to\beta e^{i\phi} $$ your state in Hilbert space remains unchanged.

They are equivalent. Let me first remind you that a '2D' Hilbert space are described by only 2 independent real parameters. You though it would be 4, but the normalization constraint you mentioned minus 1 out of 4, and a global phase minus another 1, so it's 4-1-1=2.

FYI, the 'global phase' means for the following transformation $$\alpha\to\alpha e^{i\phi}, \beta\to\beta e^{i\phi} $$ your state in Hilbert space remains unchanged.

The reason people would like to use Bloch sphere is that it’s more straightforward to illustrate. Also it’s able to generalize to higher spin case.