Difference between wire wound and ceramic chip inductors
The internal structure is similar to that of a multilayer ceramic capacitor: 



The wirewound type has less capacitance between windings, leading to a higher self-resonant frequency. Also lower DC resistance (so higher Q) and a higher DC current rating. The multilayer type can support higher inductance in a small package. 

Either type can be made with 'air' (actually ceramic, but it has similar magnetic properties but higher dielectric constant) or ferrite, with the former suitable for very low inductances.