quantum mechanics - Bloch Sphere vs 2D hilbert Space

For a 2 level quantum system we can represent it in two ways: in 2D Hilbert space and also in the form of a Bloch sphere

Now,

In what space does the Bloch sphere lie?

Are they equivalent? because what seems slighlty counterintuitive to me is that in a '2D' Hilbert space we are representing a point by two complex numbers which is 4 real parameters, whose magnitude I admit are related by (alpha)^2+(beta)^2=1, but in a Bloch sphere we represent it just by two real parameters, so are 2D Hilbert space and Bloch sphere equivalent representations of a two level system?

Why is one method of representation preferred over the other? in the context of quantum information processing