The mile is a unit of length in the imperial unit system with the symbol mi.
It was originally an English unit but was adopted by many countries who all had their own slight variation on definition. The English mile, however, is equal to 63,360 inches, 5280 ft or 1760 yards.
The UK still uses the mile on all roadway signs rather than the kilometre; its SI "equivalent". It used to have the symbol 'm' until SI was established and it was changed to avoid confusion with the unit metre.
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.