They both come from the Greek word pneumon, which meant either breath or spirit. For that matter spirit means either breath or spirit. It comes from the Latin spiro, or breathe, and still survives in respire and respiratory diseases, such as pneumonia.
The fun thing is that when you look up pneumatic in the OED, the first definition is this:
Belonging or relating to spirit or spiritual existence; spiritual.
Chiefly in the context of New Testament theology.
That's from 1624, and pneumatic doesn't get its engineering meaning until thirty years later. One of the citations (from 1899) has:
The Gospel of St. John—the pneumatic gospel, as it was called, or gospel of religious genius.
Pneumatic seems to have kept its spiritual meaning and, in the nineteenth century, Jeremy Bentham invented the lovely word pneumatico-hedonistics which means:
...the branch of study that deals with spiritual or mental pleasures (as distinguished from those of the body)
And it turns out that I am much more devoted to somatico-hedonistics: "those branches of art and science which, as above, have for their objects those modifications of pleasure, which have the body for their seat."
But the funnest pneumo-word in the OED is this:
Pneumonoultramicroscopicsilicovolcanoconiosis n. (also pneumonoultramicroscopicsilicovolcanokoniosis) [compare pneumonoconiosis n.] a word invented (prob. by Everett M. Smith (born 1894), president of the National Puzzlers' League in 1935) in imitation of polysyllabic medical terms, alleged to mean ‘a lung disease caused by the inhalation of very fine sand and ash dust’ but occurring only as an instance of a very long word.
The Inky Fool goes for a spin