"Shanty" & "chantey" are mere spelling variants. The first reflects unambiguously the usual pronunciation; the second reflects another possible pronunciation and also the best guess at the origin of the word (French chantez).

The American Heritage Dictionary gives chantey first, and also lists chanty, shantey, and shanty. The Oxford English Dictionary gives shanty first, and chant(e)y as variants. Thus, it is possible that the sh- spelling is more common in Britain & the ch- one in the US; but there is plenty of variation on both sides of the Atlantic.

Both dictionaries give the sh- pronunciation first, and the ch- one second.

It's a matter of taste. The existence of the unrelated word "shanty" = rude dwelling hut, which is more common & thus more distracting in the US & Canada, causes me to prefer the spelling "chantey", tho like most people I use the sh- pronunciation.