Generally the AnimationPlayer just contains the scene's animations, and you save it along with the scene. You can run into problems if you move the AnimationPlayer around and the track names no longer reflect the node paths correctly. You'd have to go in and manually fix these track names.
Saving off individual animations as resources is for more niche things. Maybe such as copying specific ones to other AnimationPlayer that has the same scene structure. Swapping them in and out using code at runtime, etc. I think you'd only want to do it if you can think of a specific reason why you'd need to do it.