Why should I use LaTeX?
Superficially, one of the advantages of LaTeX over other more traditional systems (e.g. Word or OpenOffice) is the high typographical quality of the documents that you'll be able to produce. This is particularly true for documents that are heavy on mathematics, but documents for any other area could also take advantage of these qualities.
A less obvious advantage, but much more important, is that LaTeX allows you to clearly separate the content from the format of your document. As a writer (scientist, researcher or not), this gives you the opportunity to focus on the “what”, the creative part of your work, rather than the “how” is it going to look printed out in paper (that is the work of LaTeX document class designers).
Now, you shouldn't use LaTeX if
You don't have time to learn it. Unlike most other point&click systems, LaTeX does take some time to learn. There are of course many guides and tutorials that can help you with this, but don't try to learn LaTeX if you have, say, less than 24 hours to prepare a manuscript.
Your document is already written. Say, if you have already written your thesis in Word, there isn't much point in trying to “convert” your document to LaTeX. You can do it, but the results won't be pretty. LaTeX isn’t just another document type to “Save-As”, it's a complete system to help you write those documents.
What you care about is the design of the document. If you do care about creating your own designs for your documents (rather than the content), LaTeX is perhaps not the best system for you. There are a number of packages (perhaps most notably memoir) that allow you to customize the look of your document, but things are not always straightforward. Having said that, if you are a designer, of course we would welcome your help in designing new document classes and templates!
I don't intend this to be a complete answer to your question, but I don't believe anyone else has yet brought up the topic of longevity.
TeX has been around for over thirty years, and the underlying language hasn't changed very much in that time. Has anyone ever tried using Word 2007 to load a file that was originally written in Word '97? Even if the file imports properly, chances are some of the page/line breaks are going to be off, possibly skewing the entire layout.
What if you had decided to write your document in Lotus Word Pro back in 1990? Would you be able to view/edit that document today? These problems almost never occur with (La)TeX.
A document that is typeset in (La)TeX today is likely to look exactly the same when you re-typeset it 10, 20, or 30 years down the road. Since TeX is stored in a human-readable plain text file, you also have the knowledge that you will always be able to edit the file in the future.
Regarding the benefits of LaTeX, some in short:
LaTeX provides very high quality and is extremely customizable.
It's extremely stable, no matter how complex the documents are.
It's free and Open Source, we can study and improve everything as we do on this site.
LaTeX is portable concerning its implementation, your document source and its output -- all is cross-platform.
It provides a logical approach to create documents instead of a physical, enhancing consistency.
Your document is safe because the file format is open and there's no virus threat.
I elaborated these points in this article: Getting Started with LaTeX.
If you have questions to any of these points, don't hesitate to ask.