X-ray diffraction ends up being a beautiful application of Fourier theory.
You are measuring the $k$-space (reciprocal lattice space) points with the x-ray diffraction. The Fourier transform of these points (provided you measured enough of them) gives you exactly the Bravais lattice.
The Bravais lattice is the crystal stucture! You then have to look at the Bravais lattice and construct a Wigner-Seitz unit cell, then that construction will determine your crystal type.
Alternatively, you could analyze the reciprocal lattice space "unit cell," find what it looks like, then Fourier transform that cell to get the Bravais lattice cell type. The Fourier transforms here is on an infinite periodic array, if that wasn't clear.
The point arrangement actually determines the FCC or BCC type...I have included a diagram of the two different types (FCC and BCC) shown in the reciprocal lattice space below, taken from Introduction to the Physics
of Electrons in Solids by Henri Alloul. You can see that they are not the same...