Quality and Aesthetics
Simply, documents produced using LaTeX just look better. There are 2 main reasons: typography and the ability to create complex designs.
Text in LaTeX documents looks more refined and polished because LaTeX uses iterative typesetting algorithms which determine the optimal layout of text and floating elements (such as figures and tables) based on many typographical rules. Word processors are not written to typeset documents (i.e. determine the optimum position of characters and words) on the fly.
The reason that the design and layout of LaTeX documents can be beautiful is that LaTeX is capable of creating virtually any design by making use of layering and absolute positioning of elements, along with the ability to output text in columns and boxes. In this sense, LaTeX is as capable as a professional design tool like Adobe InDesign, but without the cost.
A lively community of expansions to LaTeX (called packages) allows virtually any modification and type of document to be created. This means that LaTeX can create everything from business cards to books to presentations. Advanced built-in referencing also allows LaTeX to easily handle very large and complex documents with many cross-references, such as theses.
Price
Free and works on all operating systems.
Microsoft Word, Apple's Pages and Google Docs are popular word processors that make creating documents easy and interactive. All cost money and/or your privacy, have compatibility issues between operating systems and either aren't available on all operating systems or require additional licenses. There are free open source alternatives such as LibreOffice, but even these have compatibility issues.
LaTeX is free and there are a multitude of text editors to choose from for each operating system. If you create a LaTeX .tex file at work on Windows and resume working on it at home on your Mac or Linux computer, the file will compile the same regardless. Since LaTeX and individual packages do not get updated very frequently, you are unlikely to run into incompatibility issues due to different versions.
Another consideration is bibliography management. For the word processors that are capable of including bibliographies automatically, this software is usually an extra expense (e.g. EndNote for Microsoft Word). The LaTeX implementation of a bibliography manager (called BibTeX) uses any number of free reference managers to store your papers and saves your library as a .bib file which is read by LaTeX and output in a citation style of your choosing.
Editing, Versioning, Automation and Outputs
LaTeX code in a .tex file is just a text document. This means it can be opened and edited in any simple text editor on any version of any operating system. It will not crash or corrupt for seemingly no reason. Many people feel this is liberating and enjoy being in control of their documents. Word processors can seem slow and bloated but few computers will find a text editor a daunting application to run.
Software developers commonly use source control systems to track changes to files over time. LaTeX is very amenable to using Git to track changes to your documents, something that is difficult to do for documents from a Word Processor. Git also makes it easy to collaborate with other authors and maintain multiple versions of your documents using its standard features.
The benefits of plain text also mean that LaTeX is very easy to automate for business use. Simple regular expressions, when paired with a good template, can enable a business to produce documents on demand and in large numbers with no extra costs.
The output from LaTeX by default is PDF. The PDF format is used extensively worldwide and is viewable on any operating systems without formatting differences. Along with PDF, you can output to PostScript, RTF, HTML, PNG, TIFF and others through the use of other free tools.
Focus on Content
LaTeX separates content and styling/design, and this is especially true when using templates. What you see in your .tex file is not what you get as the output. While this makes things more confusing for those that aren't accustomed to compiling documents, it is actually liberating in that the structure of the document is set using rules and the content falls into place by following the rules when the document is compiled. This ensures consistency since the formatting is handled separately and en masse.
Document content can be written without the distraction of seeing unwanted changes appearing as you type, and can be compiled when you are ready to take care of formatting. For example, if a footnote is required, it is simply written into a \footnote{} command in-line with the text and the writing can immediately proceed without worrying whether the footnote has been positioned correctly.
Time Investment
There is no doubt that LaTeX requires an investment of time in order to learn how it works and how to use it. The key word here, however, is investment. When using a word processor, time spent trying to fix a formatting problem is essentially time wasted; it is unlikely the same problem will arise again. In LaTeX, once you figure out how to make a formatting change, you now know how to do this forever and it becomes quite natural given some repetition. Each new task builds on top of previous knowledge to make the process easier, and there is extensive help available at places like Stack Exchange to answer any questions you might have.
Longevity
Documents written 20 years ago in LaTeX are almost guaranteed to compile today. Compare this with a document produced using a word processor from 20 years ago, it is likely the software no longer exists and if it does, it has undergone so many changes that it may not display the document correctly.
Downsides of LaTeX
Learning Curve
For those with little experience beyond point and click interfaces on a computer, LaTeX has a significant learning curve to efficiently create documents. This may take differing amounts of time to surmount based on previous exposure to programming or command line operations. Time is often a luxury in a professional or academic setting, so it is one of the aims of this website to help ease new LaTeX users into the system by providing ready-made templates that can be quickly filled in by a novice.
Collaboration
Since LaTeX outputs read-only PDF files and most people aren't comfortable editing LaTeX document code, it can be difficult collaborating with others on a document. For proofreading purposes, PDFs can be annotated with notes, highlights and lines to cross out text. macOS has built-in PDF annotation in Preview and Adobe Acrobat is available for Windows, but this solution can be slightly more cumbersome than the automatic 'Track Changes' implemented in Microsoft Word. Another option is to print the document and make annotations by hand. If a collaborator requests a document as a Word file, this represents further problems since both a PDF and LaTeX code are not easily translated to plain text and formatting will not be retained. Overleaf is a web platform that is tailored to collaboration on LaTeX documents so this may be a good solution if this is an issue for you.
Lack of Immediate Preview
In a word processor, when a change is made, it is immediately obvious what has been changed. In LaTeX, every change can only be seen after the document is compiled. This means LaTeX does not provide the immediate feedback that a word processor does, which adds another layer between the user and the document. This is both a blessing and a curse in that you are able to write without worrying about the look of the document and compile at a time when you are ready to see the document and add formatting, but when you do compile, you may see cryptic errors that can be difficult to diagnose or formatting that you need to manually adjust.
OK, this sounds interesting, how do I start using LaTeX?
