crystal = lattice + motif
The single most important idea in crystallography is not really numbers: crystal = lattice + motif. Take the geometric grid of points (the lattice), decorate every point with the same atom-group (the motif), and you have built a crystal. Wallpaper = the grid of positions + the repeated picture.
Formally, the crystal is the convolution of the lattice with the motif: place a copy of the motif at every lattice point. Two crystals can share the very same lattice yet have utterly different motifs (copper and rock salt are both built on a face-centered cubic lattice, but their motifs are nothing alike), and the same motif can be hung on different lattices.
This separation is exactly why diffraction works: the lattice controls where the diffraction spots appear (peak positions, the unit-cell geometry), while the motif controls how strong each spot is (intensities, the structure factor). Positions give you the cell; intensities give you the contents. Keeping lattice and motif distinct is the mental discipline the whole of crystallography rests on.
Copper: FCC lattice, motif = one Cu atom, giving 4 atoms per conventional cell. Rock salt (NaCl): the same FCC lattice, but motif = one Na plus one Cl, giving 4 Na + 4 Cl per cell. Same lattice, different motif, completely different material.
Two very different crystals built on one and the same FCC lattice.
Because diffraction separates the two, you can determine the cell (the lattice) from peak positions long before you know the atoms (the motif), which come from the harder analysis of peak intensities.