(La)TeX is a markup language, meaning that you code in a regular text file both the content and the layout of the document that is later computed by LaTeX.1
Thus, I think that there are two situations when using LaTeX is worth it. I also lists two benefits of using a markup language when producing documents. Note that my points would apply to most markup languages — previous answers are good resources to explain how is LaTeX superior to such other languages.
1: For the sake of simplicity, I've used LaTeX throughout the answer. However, I think that my answer applies to all *TeX flavours.
Two situations when using LaTeX is worth it
We have seen that LaTeX is a markup language. This means that you code two things: the content on one side, and the look, or layout, (ideally) in another part of your document. Moreover, you should remember that LaTeX is a computer language: i.e., it does exactly what you tell it to do. This means that you can control all the handles (very high — and fairly easy — customization potential). But also that it is very dumb and will not guess if you won't tell it to do it (you must explicit everything that you want different from the default case).
Hence, there are two situations where — in my opinion — it is worth using LaTeX compared to a word processing software:
- when you want to focus on content only,
- when you want to produce a heavily customized document.
Note that the graph above do not take into consideration the initial time and effort investment required to achieve a basic mastery of either LaTeX or a word processor program. I think it is not controversial to claim that learning how to use a WYSIWYG word processor is notably quicker than learning LaTeX.
1: Focus on content only
In such case, you want to code content only and don't want to bother with customizing the look: you rely on the default layout of the class you are using. In other words, you spare yourself one half of the job (defining the layout). You will then produce a quality document with advanced features (font, math, references, etc.) without the hassle of setting all this by hand — as it would be the case when using a text processing document (even if style sheets are getting smarter, the features-and-quality-over-hassle ratio remains in favour of LaTeX).
Note that in such situation, I would be tempted to recommend using an even more streamlined "interface", such as multimarkdown, in order to limit the handles you can activate and force you to focus on content only.
You may be tempted to create your own customized class: it then falls in the second situation.
2: Heavily (or finely) customize your document
The second situation, is when you want to customize your document a lot. Add a tons of complex features (list of acronyms, index, main bibliography plus one per chapter, flipbook animation, etc.). Have an OCD with correct kerning and spacing. Replicate a cool layout you have seen on TeX.SE. In such situation, achieving your desired result will be easier and quicker using LaTeX — because you have a direct access to every single parameter in the document processing (provided that you invest enough time learning how to do it). In other word, LaTeX is not a black box — unlike word processors that use black magic to guess and ease 99% of your work, but won't let you access to very advanced settings. (Have you ever thought "But why does the figure in page 17 moves to the end of the document when I add a coma in page 13?!" or "How could I make this specific feature: there is no button for it?")
Note that — even if pretty much everything can be produced using LaTeX — due to the computational nature of LaTeX, this situation is more relevant with "traditional documents" (books, reports, theses, letters, etc.); i.e., documents using a layout template. If you are producing an one-time-shot artistic book with a specific layout for each single page, I think indeed that you would be quicker using a dedicated software (such as Adobe InDesign) — as I think the WYSIWYG approach (and hidden computation of click-and-drag drawn text areas' coordinate) still remains faster.
3: Situations in between
To my opinion, in other situations, using LaTeX is not worth it: you will invest a lot of time for explicitly coding every single deviation from the default layout when a quick and dirty manual "hack" in your word processor may certainly not be robust in the long run yet do the job and take a couple of seconds only.
Of course, this is simplified, and in some cases, you will need a LaTeX feature so bad that investing time will be worth it — even if you're not customizing your document a lot. Moreover, learning and using LaTeX can be fun. But I would not say that every one should use it every time: to me, there are cases when it is just not worth it. Even when you know LaTeX already: I think that I can consider myself as a power user and yet there are still documents I prefer to write using a word processor. It is when I don't find default layout sufficient, yet coding a dedicated one would take longer than doing it with the mouse using OpenOffice/MS Word. I'm thinking of creating my own customized classes, though — yet it is still on my to-do list.
One should also note that customization does not mean complexity: you can very easily produce complex documents without customization (see, e.g., KOMA script classes that comes with tons of embedded features by default). And very basic documents with a hint of you personal flavour or a specific look way require a lot of customization.
Two features of LaTeX that makes it worth it
Previous part examines when using LaTeX is worth it. I'd like to add two more features due to the nature of LaTeX — that is a markup language.
Using version control
Version control (git, GNU Bazaar, CVS, Subversion… you name it) is a powerful system: it's a save button combined with a time machine. More detailed explanations are beyond the scope of this answer. But just know that it is a very powerful tool that a lot of people use and serves to easily reverting to an earlier version of your document without storing lots of copies of it and to enable real-time collaboration on a single document (two persons modifying the same document at the same time). Long story short, version control works with plain text files (i.e. source code) — i.e., with LaTeX files but not word processors' ones. This is a highly valuable feature and word processors' ersatzes are not as powerful.
Automatic document generation
Because LaTeX needs a text file only to create a document, it is very easy to create scripts that automatically generate documents using a database: weather report, train timetables, catalogue, etc. It is very easy to pipe such results into a predefined layout and let the computer do the rest. This way, you truly benefit from a major skill of a computer: the ability to do repetitive tasks quickly.
This is not focusing on LaTeX strictly, but another feature of a LaTeX (by means of packages — mostly TikZ) I value a lot is the ability to code/compute your graphs and vector figures. That means that you can pipe some data to your document and create self-updating graphs (see "automatic document generation" above). But you can also draw pretty complex graphs/figures in your document, meaning that your source code is self sufficient to produce the document (no more "where is the editable file for this graph?"), and that you can use your documents settings (length, colours) in your graph to create top quality documents.