There's more than EM and EN spaces (see below). These are used in advanced typesetting to create optical adjustments between elements or to avoid using repeated spaces.
So instead of typing Space multiple times to move something, you can use an EM space to have a clean file with no repeated spaces. In extreme cases you can use an EM space with a blown up Horizontal Scale (500%) to push some text really far away on the same line without using repeated spaces.
Other times they're useful to adjust spacing around EM or EN dashes. Also useful in titles, tables, chart labels, separation between left and right aligned wording, indents and so on. Also, i've seen legal documents where each sentence was followed by an EN space instead of a regular space to separate ideas better inside the same paragraph.
Later edit: you can also use these spaces for some really smart alignment tricks like this one here: https://graphicdesign.stackexchange.com/a/115602/62949