I've always been curious: Pretend I'm totally ignorant, and have no idea what need a typesetting tool fulfills. Someone who has used and been happy with Word / Open Office for years to write documents. What does LaTeX/InDesign solve for me, besides "nice looking math" that I'm not getting out of basic word processors? How do I know which one I need? I honestly have no clue. If I were to start writing a book tomorrow, I'd just open Word. Why not? What am I getting when I invest either the time to learn LaTeX or the money to buy InDesign?
For writing a non-fiction book, you should probably use Word or something similar. Just make sure to avoid any formatting tools, only use semantic markup. Word is a pretty decent authoring tool. The end result can then be imported to a typesetting tool for formatting.
InDesign is used when you need precise control of the output. For instance controlling exactly how figures are placed, you might more advanced control of how text flows between boxes, etc. It also has tools for very precise control of how the type is set, how big space should there be between letters and words, how should the right edge of columns look, should there be different number of columns on different pages, etc.
Some of those things can be somewhat managed in Word, but you'll have to fight a lot of the automatic stuff, really not worth it if you are a full time design professional, much cheaper then to buy an expensive InDesign license.
Latex is pretty good at having sane defaults. This was a bigger issue back in the days when the defaults of Word were frankly terrible. Today it's to a large extent about style choice. If you publish in an area where Latex dominates, the Latex styling will make your document appear as more serious. Latex also generally uses a more advanced type setting engine, for example it might join "fi" with ligatures etc. This can also improve the look of the document.
Some people like the fact that you can manage Latex code as raw text. For instance using a VCS to manage version history. Word has some built in version management functionality, but it's quite clunky compared to Git.
Personally I gave up on using Latex after my first master thesis and tend to use Word. I get too caught up in the formatting when I use Latex. Maybe it's too much power to handle for me?
