Do we need both? Looking at the questions list, I can see arguments for keeping both, as there is at least one question there where bone is valid but skeleton isn't. However, the two are very closely related and I'm wondering if we couldn't use one tag for both.
Yes, I would say we need both. Because bone can be bone in general
(including things such as nails), skeleton relates more towards specific things such as Exoskeletons, Skeletal Structure, and undead skeletal creatures.
As such, we can have things that require Bone, but not Skeleton. (Eg: Question asking about bone claws)
Also, as Michael has noted, you can have skeletons that aren't bone.