What is the difference between atomic structure and crystal structure?

Atomic structure applies to just that, atoms in isolation. Witness looking at atomic spectra using laboratory discharge tubes to ionise low pressure gases of atoms of the elements you want to examine for energy level structure (sodium and the yellow D line "doublet"). Crystal structure applies to vast collections of atoms/molecules/ions in complicated geometrical arrays which form lattices. Simple cubic, body centred cubic, face centred cubic being a few "simple" examples.