What are the advantages or disadvantages of using LaTeX for writing scientific publications

LaTeX is better at:

• Dealing with mathematical notation. Layout and entry are generally easier using LaTeX than some other sort of equation editor.
• Consistent handling of intra-document references and bibliography. As of a couple of years ago the major WYSIWYG editors still had problems with re-numbering cross-references and bibliography items. This is never a problem with BibTeX or LaTeX.
• Separation of content and style. In principle this means that you can write your document without caring how it is formatted, and at the end of the day wrap it in the style-file provided by the journal publisher before submission to conform to the house style. In practice some of the journal publishers demand special formatting commands that partially moots this process. Furthermore recent versions of Word and LibreOffice Writer, when properly used, should be able to keep track of various levels of section heading separate from the body text, and apply uniform styling to each level. The gap is somewhat closing.
• Tables and illustrations. With PSTricks or TikZ, one can produce high quality illustrations within the document (though the learning curve is a bit steep there). And I've found LaTeX to be better at preparing complex tables.

WYSIWYG (especially Word and Writer) is better at:

• Collaborative editing. Without using an online site for collaborative LaTeX editing (such as ShareLaTeX), working collaboratively on a LaTeX file ideally requires some sort of separate revision control software. Word and Writer have very good comments/annotations and edit-tracking features. When a large number of authors are commenting on the writing of one file, this can be very useful.
• Spell check. Admittedly most text editors one uses to edit TeX files also do spell check. But this is generally conveniently built into WYSIWYG editors.
• Compatibility. Unless you work in mathematics, computer science, or physics (and sometimes even if you work in those fields), it is more likely that your collaborators will know what to do with a Word or Writer file than a LaTeX file.
• Minimum barrier to entry. If you just care about getting the ideas down on paper, you can use a WYSIWYG editor just like a typewriter. Sure, it may be tedious to fix the formatting later on, compared to LaTeX where one need to first figure out how to setup a bare-minimum file before using, it may be an attractive point.

A wash:

• Most journals provide templates for both Word and LaTeX, so there's no real winner there.

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Suggestion: if you want to convince someone to start using LaTeX, start them out first in one of the WYSIWYG environments designed for LaTeX first (for example LyX). This will help somewhat to ameliorate the scary entry barrier.

Word

Advantages:

• Minimal learning curve
• WYSIWYG
• Ridiculously high install base; almost guaranteed that anyone can read/edit your file without modifying anything
• Easy-to-use reviewing tools (view changes, add comments, etc)

Disadvantages:

• Layout can be a real bear to get correct
• Math is difficult, slow, and often ugly
• Included bibliography editor is virtually unusable for most scientific writing; you'll need to buy a third-party solution to manage your bibliography

LaTeX

Advantages:

• It's just text; anyone can edit your file
• You don't have to worry about layout, it's all automatic. Just put the words down there and you're golden
• Produces aesthetically beautiful documents
• Easy to use math, symbols, etc
• Once learned, much faster and more intuitive (e.g., \label and \ref for referencing, as opposed to Word's fairly kludgy "Cross-referencing" window which requires way too many clicks to insert a single reference)

Disadvantages:

• Fairly steep learning curve
• Collaborators unfamiliar with LaTeX will have difficulty reviewing your manuscripts
• Many features require libraries, which you have to find/be made aware of (view changes, etc)
• Layout changes are difficult (i.e., will require time for you to hunt down solution and implement it)

I am currently studying for a PhD (engineering, almost done) and I supervised some student projects (diploma thesis and so on). Therefore I experienced both the typical Word and LaTeX workflow.

Most of the points are already mentioned in the very good other answers. I just want to add some general comments. Since I am not a geek I think that I can give roughly an average opinion.

The Typical Word User

• Almost every student was complaining about mysterious behaviour like disappearing pictures or lost formatting.
• In addition, it happened quite often that large documents get somehow corrupted and it wasn't possible to edit/open it again. Then the students had to fall back on one of their backups.
• The larger the document the harder it is to keep the typography consistent. Because it is so easy to edit the text manually (this one bold, that one italic and so on) the students usually pay little attention to the systematic use of style sheets (don't know if that's the correct English term).
• Dealing with complex mathematical expressions is painful.
• Many of the settings are done using some menu entry and so it is very hard to document the settings or to make comments on how or why someone did something.
• Only one out of 100 students knows that you can work with vector graphics (in this case EPS) in Word. So most of the figures are ugly.
• In the last days of the thesis it is an advantage in Word that you can -- in the case of an emergency -- just draw anything everywhere in order to meet specific goals.
• It is hard to reuse content in Word -- I mean reuse it that way, that a change at one instance affects all instances.
• ...

It is not important whether all this happened because of a unprofessional use of Word -- the point is, that it happens.

The Typical LaTeX User

• You need a friend!: It is almost suicide to start using LaTeX on your own when you start writing your last and most important thesis at university. This can only work if you have a friend or colleague (like me) who can give you a template and a crash course.
• Almost all students who decided to use LaTeX reported that they had fun using it! I think the main reason is that the documents look very professional without doing much.
• In addition, students find it "cool" that the output is directly a PDF with hyperlinks and so on. Most of the normal WORD users don't create cool PDFs.
• Students like the fact that they can easily input PDF files like datasheets to the appendix. This way they even appear in the table of content (TOC). I have never seen a Word document with a detailed TOC for the appendix. This is especially important in the field of engineering.
• Using/creating bibliographies is not fun. But after they figure it out it works fine.
• The concept of using a distribution like MiKTeX or TeXLive is new to most of the students and confusing at the beginning. In addition they sometimes can't distinguish between LaTeX itself and the LaTeX text editor (e. g. Texmaker). Therefore I made a video tutorial for that ;): http://www.youtube.com/watch?v=RL15I-6NQFQ