Well, this exchange got me looking stuff up and trying things out. (TL;DR)
What's the difference between Median and Center?
- Median is the vector-average point of the current selection. In Object mode, that's the vector-average of the selected object origins - if you have only one object selected, that's its own pivot. In Edit mode, that's the vector-average of selected vertices, whether selected as vertices, edges, or faces.
- Center refers to the current Transform center, as set in the Transform Pivot Point dropdown. For example: if in Edit Mode, that's 'Bounding Box', the snap source will be the center of the bounding box of the verts in the object's orientation, not the average position of the vertices. If it's '3D Cursor', the snap source is the position of the 3D cursor with respect to the selection before the move started, which can be very useful. If you can put the 3D cursor at any point with respect to your geometry, (often using ShiftS,) you can use its location as your Snap source.
What I don't get is why, in Object Mode, 'Center'+'Bounding Box' still snaps the object's pivot, not the center of the object's bounding box.
There's also the A feature, which lets you select the weighted average of multiple snap targets. If Snap is locked on with the magnet, you can hit A to add the current hover-target to an accumulation to be averaged. This means, for example, you can hit A twice at one end of an edge, and once at the other, to snap to a point 2/3 of the way down it.