In a vertical mill the cutter axis is vertical and the cutting action takes place on the end or side of the cutter. In a horizontal mill the cutter rotates in the horizontal plane and the cutting action is on the circumference or face of the cutter.
The other key aspect of a mill is that the workpiece can be moved and accurately positioned in three dimensions relative to the cutter.
Therefore a CNC router IS a (limited) form of vertical mill.
Here is a useful description of various cutting processes...