This question is often heard and usually overseen.
When arising, it is custom to put it aside with something like this:
" Sound and Audio have the same meaning " (with an obvious tone)
But wait, this can't be it!
The words are not same, they have divergent origins.
I had to dug it up.
Even if a sound designer will say that he has the same job as an audio designer. The vocabulary difference of 'audio' versus 'sound' can open virulent debate.
With some good sense, research and reading, it is easy to understand that Audio and Sound have each their own path of implied values!
Here are my version of the definitions:
Audio comes from the Latin 'Audire' which means 'Hear'.
The word is used only since the beginning of the 20th century, when technology has permitted the capture, reproduction and manipulation of sound. In fact 'audio' describes anything that we can be heard but always through the relation of technology (produced, recorded or processed) implicating electricity, electronic or computers. This could also be referred to as "sound signal".
Sound on the other side is a more generic term that has been used for millennia.
From the Latin 'Sonus', a sound is described as a vibration through the air that produces a physical reaction when reaching the human's ear.
Hence sound encompasses anything that we say audible.
All audio are sounds, but not all sounds are audio!
For example, if I see a bird singing in a tree, I won't say 'it is producing an audio'. Instead we say 'it is producing sounds'.
On the other hand, a recording of a bird song is said to be audio as well as a sound.
So to wrap it up,
Sound is a broad chameleon notion whereas audio is a more specific term that relates to our technological knowledges.
Once we know the definitions, the distinction is pretty clear.
Let's not get confused and call a spade a spade!