What properties other than Henry and Current ratings should be taken into account when replacing inductors
Resonant frequency is the point where the inductor starts to behave like a capacitor so ideally you should find a replacement with a resonant frequency not less than the current device. However, if the fact that the inductor resonates is useful to the circuit this can be very bad advice. We need a circuit to tell!

Q factor is pretty much the same as resonance - higher is usually better BUT, like resonance, the circuit may be relying on a Q factor that is not that great.

DC resistance - normally lower is better in general and this also means Q factor is improved too but the same small print applies. No circuit, no can tell!

Probably the most reliable part of the answer is the core material. If all other requirements are met the only disadvantage I can see with ferrite is that it's permeability changes with operating temperature and this may cause a problem. Material specifications for both core materials are the best way to judge.

Sorry, it's not an easy answer; a circuit analysis to understand what the inductor does and how temperature may affect performance is a basic requirement for judging this one.