The mil is a unit of length in the imperial unit system and uses the symbol mil (or sometimes thou).
A mil in one one-thousandth of an inch (0.001 inches) or 25.4 μm.
The mil is still used in the US to measure thickness of various materials.
The em is a unit of length in the field of typography and uses the symbol em.
It is a "relative" unit and equals the currently specified point size. For example, 1 em in a 24-point typeface is equal to 24 points. This allows a typographer to specify text size relative to a base figure.
In metal type, the em was equal to the line height of the metal body. This meant the physical size of the letter could not exceed the em. In digital type, the em is a grid of arbitrary resolution so can be scaled to any point size for screen or print.