Compiling: umbrella term for transforming source code in one 
language into another. In practice it's mostly used to describe 
the process of transforming source code into a lower level language
(e.g. C code to a binary executable (machine code))

Transpiling: also known as source-to-source compiling. That is 
transforming source code in one language into a language of the same
level or different version of that language (e.g. Emscripten transpiles
C/C++ to JS and Babel transpiles ES6+ to ES5)