A metric space is homogeneous if for any two points there is a global isometry that maps one into the other. It is locally homogeneous if any two points have isometric neighborhoods, i.e. the space 'looks the same' near them. Take open flat disk, it is clearly locally homogeneous, but there is no global isometry that maps its center to any other point. However, the disk is incomplete near the boundary, and if we complete it boundary points will no longer 'look the same' as interior ones.
Can complete connected Riemannian manifold be locally homogeneous but not homogeneous? How about closed one? I suspect yes, but I can not think of any examples.
In cosmology locally homogeneous is usually just called homogeneous, but I wonder if this is in line with mathematical usage even for 'nice' spaces.