You will get different BH Curve for different air gaps,

Advantage and disadvantage depend on application, sometime low gap is advantageous and some time high air gap.

Just see below curve,

In general:

A non-distributed air-gap (as in stacked steel laminations) will result in high leakage. A distributed air-gap (as in, for example, MPP torroids, will have low leakage.

An air gap will cause a lower inductance since the effective permeability of the path with an air gap will be approximately

μ_core/(1 + (μ_coreA_core/L_core))

where A is the cross-sectional area of the core and L is the effective length of the magnetic path. Sometimes, particularly in designing inductors for tuned circuits, it is desirable to deliberately have a lower permeability (you may need lots of turns because of the voltage, yet want a low inductance).

Another advantage of an air gap (whether distributed or non-distributed) is that you can now have a dc current in the coil without saturating the core.