The major selling point to me, is that I can write the content I want, in a rather straightforward manner, and then just apply whatever formatting is necessary. It makes a lot of sense in the context of, say, university, where you're taking multiple classes, which each use different formatting guidelines, but those formatting guidelines stay the same for the duration of the course. I could just figure out (either using existing formats, or by hand-rolling my own) a document type for each class, and then never think about formatting again, whereas with a word processor, you've got to keep in mind the arcane series of steps necessary format your document in whatever style you require, and then pray that any modifications you make don't break the formatting.
The other place it's useful is heavily typeset documents, especially those subject to somewhat frequent modification, like a resume.
Using a word processor (MS Word, Apple Pages, LibreOffice Writer) is nearly as bad as using LaTeX (in some cases, worse).
Using a plain-text format like Markdown, ReStructuredText or AsciiDoc is likely better in almost all cases.
Plain text formats like Markdown work great if you don't care at all about the specifics of the final formatting. Tools like LaTeX are for when you care deeply that the resulting document follows a certain format, but don't want to think about that formatting while writing.