Since this decision has not been resolved so far, I took the liberty to research everything and make an opinion myself.
note: this is — all in all — my own; humble opinion. If you have any opposition, please kindly let me know. I will be all ears.
While scabbard and sheath are pretty much synonymous; used interchangeably, there are some differences as follow:
properties: scabbards are usually made out of harder, heavier materials, such as wood or metal, while sheaths are usually made out of softer, lighter, and more flexible materials, like leather or fabric.
usage: both of them can be used to protect the blade, and everything around it. However, as I mentioned above; for scabbards are usually heavier, sturdier, and more durable, they are also used for greater protection, such as for displaying. A sheath is lighter, so they're used for when you want to move around more freely, as it does not weigh you down as much.
scabbards are also generally bigger, which results in it being used for a larger, longer swords. It also makes it harder to carry many of them around, which makes a sheath more suitable fore carrying short, smaller blades such as dagger or knife, plus you can also put it on somewhere else other than your waist or back, giving higher accessibility.
...That is, if you considered both of them as two different tools.
a scabbard is — more or less — a type of sheath.
both of them can also be used for guns, for which it will be called holsters.
So, if a sheath is used for blades: that is a scabbard.
If it is used for guns: that is a holster.
Alas, a sheath will still be a sheath, no matter the tool. Axes, wands, you name it. If it covers, it covers.
Again, this is — after all — my opinion. It will be the way that I follow until further notice.
Well, if there is something I should know, please do not hesitate, for I promise not to bite.