There isn't really a difference, just an enhancement in design by including the ferromagnetic core for an inductor. An empty inductor is just a coil of wire. A ferromangetic material core inductor is a coil of wire filled with a ferromagnetic material like iron as a method to get more inductance with less wire. And there is an annoying non-ideal phenomena called magnetic hysteresis in a ferromagnetic core incuctor. This only occurs when the current is high enough to saturate the ferromagnetic material's ability to enhance magnetism.
Difference lies in their inductive reactance.
Induction with a metal core produces a far stronger force than an inductor with an air core. The metal consolidates the magnetic field lines so they are greatly intensified over just a air core inductor
