What is the difference between LaTeX and MathJax?

I'm trying to answer it in a simple way utilizing Wikipedia:

MathJax is a cross-browser JavaScript library that displays mathematical notation in web browsers, using MathML, LaTeX and ASCIIMathML markup.

That means that MathJax is a software that works in your browser on some text which is marked up using different techniques. One of them is a LaTeX-like notation. In general it is independent of usual LaTeX binaries and the most part of the markup LaTeX provides.

LaTeX on the other hand is referred to as document preparation system. It is also used as term for the markup language that's used to write the document. In general LaTeX is a) a TeX format (so the markup language) and b) an executable that processes documents in (a) into the DVI format.

The MathJaX tag wiki page has more information and links

"LaTeX" is used in two senses, the name of a document markup language with syntax like \section{Intro} and \frac{a}{b} but also the name of the program that implements the processing of that language to produce typeset output.

MathJax does not use the (la)tex program at all, it is a javascript program run in your web browser (or other javascript engines such as node.js) which renders various mathematical markup languages, notably a language that is essentially the math subset of LaTeX syntax, but also the MathML markup language which uses XML syntax.