ABC is a tune notation format. It's very good for a single line of melody plus chords, not so good for multi-part music (although an extended format is coming ...). It's mostly intended for modern, Western music, i.e. 12 semitones/octave, bars, time-signatures, etc. Chris Walshaw's abc page is a good place to start, with an explanation of the format and links to software, tune collections and ABC-specific search engines such as JC's ABC tune finder.
TeX is a mathematical typesetting system (the X is actually χ, a Greek letter chi, so it's pronounced tek, approximately). It's not WYSIWYG and takes some getting used to, but it's very powerful and extensible, and widely used by professional mathematicians. LaTeX is a standard set of extensions designed to make TeX friendlier to use. The UK TeX archive is a vast repository of all things TeX. (La)TeX normally produces files in either DVI format (device independent) or PDF format (portable document format). Both of these can be previewed and printed using a variety of packages, on a variety of platforms. On this site I've used PDF files: you might be able to view these directly in your browser, or you can use a viewer such as Adobe Acrobat, which is free.
The point of both ABC and (La)TeX, for me, is that they use source files that are human-readable, platform-independent and in a widely-publicised format. This means that you can edit them with your favourite editor (i.e. GNU Emacs) and usefully exchange them with people on alien machines via email. It also means that instead of relying on proprietary packages lots of people can write helpful programs that the system's inventor didn't have time for or hadn't thought of. For example, for music I use the following packages - all of these are free, although some are still licensed:
The full TeX system is quite large and complex, and I wouldn't recommend installing the whole thing just for music. However, there are many other packages available, including cut-down versions of TeX for music: see Chris Walshaw's abc page for a list.
This page is maintained by Thomas Bending,
and was last modified on 28 March 2016.
Comments, criticisms and suggestions are welcome. Copyright © Thomas Bending 2017.