While searching about the difference, I came across these definitions:
Compiling is the general term for taking source code written in one language
I mostly agree with tapananand answer, but...
Words are "made", so they serve a purpose. And this also changes with time.
We now tend to use transpiler to specify a compiler that translates code into some other code "more similar" to the source one, that what a compiler might do. And is used to differentiate both mostly when both of them are mentioned in the same context (again mostly implying that a transpile language will have to get compiled at least once more)
So everything is very subjective. At the time of this writing:
So transpile is at this time very rarely used anymore, and only to tell two compilers apart.
It will probably dissapear as concept, since compilation is much more complicated than that (same/higher/lower language, version, etc), and the word doesn't seem to be useful anymore ("transpilers" are now ubiquitous)