Animation looks different on identical rigs

I've got two files: an FBX of a character model with rigged skeleton, and a second FBX containing just an animation for that character, using (I'm almost positive) the same exact rigged skeleton from the first FBX. I'm using Humanoid animation. If I use the "Copy From Other Avatar" option on the animation and point it to the original rig, and use that original avatar on the Animator component, the animation is slightly off in several places. If I change it to "Create From This Model" and set the Animator to use this avatar from the animaiton, it looks correct. Can anyone tell me why this is? Shouldn't it look identical if the animation fbx is using the same character rig as the original fbx?