Air Core Transformer & Ferrite Core Transformer
The main reasons ferrite cores are used are: -


They contain the magnetic field better than air preventing coupling to other circuits
Higher value inductances can be made with much less copper - trading off copper loss with core loss gives the designer more options.
Coupling of two or more windings is more effective (should this be required)
Ferrite solutions (when applicable) are usually physically smaller than air.


The main reasons ferrite cores are not used are: -


Permeability drifts with temperature hence air cored inductors are more stable in some applications
Air cores don't saturate and produce 3rd order harmonics like ferrite. Useful, unless you are designing a saturable reactor of course.
For tuned circuits in the VHF and above, ferrites are too lossy in some applications.


These are the main pros and cons.