That's a tricky (trick?) question. Where to start?
MathML is MathJax's internal format (essentially anyway) so anything that can be done in MathJax is done through our MathML support, cf
http://docs.mathjax.org/en/latest/mathml.html. While MathML is quite good for such an internal purpose, it can be difficult to create. It's rarely written manually (much like HTML or CSS) and tools can have trouble producing high-quality MathML (converters can fail, editors might produce overcomplicated MathML). MathML is the dominant format used in professional publishing workflows and thus comes with a rich toolchain out of XML-land.
MathJax's LaTeX-like input provides a faithful implementation of the most common math-mode LaTeX commands as well as other standard packages and a few non-standard features, cf.
http://docs.mathjax.org/en/latest/tex.html. LaTeX is much easier to author by hand than MathML and provides the typical LaTeX advantages such as custom macros (for even easier authoring). It also has the benefit of a large community thanks to the wide adoption of TeX as a programming language for print layout in academic writing. LaTeX is probably the most popular format when people have a choice, so MathJax's TeX-like input has a wide community out there. From a real TeX perspective, MathJax restricts LaTeX input to math-mode since it converts internally into MathML. Due to LaTeX's print heritage, some constructions are hard to do (e.g., equal-width columns are trivial in MathML but not doable with the default LaTeX macros).
AsciiMath is a lightweight markup language designed to convert well to MathML. I sometimes like comparing it to markdown -- not as powerful but much more sensible to write. It does not have the expressive power of MathML but it is very easy to learn because it was designed by Peter Jipsen specifically for high-school- and college-level students. It is less frequently used but if it's expressive power is sufficient, I tend to recommend it.
In summary, MathML is MathJax's internal format so anything you can do with MathJax you can do with its MathML input. LaTeX is virtually as powerful (with some edge cases), is easier to author by hand, and has a large community both from real TeX-land and MathJax's community. AsciiMath is the little brother of both MathML and LaTeX and provides a good compromise between expressive strength and human readability.
If you look beyond MathJax there are even more options, of course.
I hope this helps a little.
Best regards,
Peter.