"Pox, when used without an epithet, as in imprecations, formerly signified smallpox; but it now signifies syphilis.
Webster's Dictionary, 1913
While the pox was indeed the common name for syphilis from the 1500s and on into the 1800s, there is a bit more to the story than may be apparent to the modern reader.
A pox was originally a pock; the Old English pocc referred to a pustule, a pus-filled swelling on the skin, and to the mark left after its passing. The plural of pocc was pockes, eventually to become simply pox. There were many types of pox, but for a long time the most fearsome was smallpox -- although at that point there was nothing small about it, and it was simply The Pox.
In the late 1400s a new pox appeared on the scene, perhaps brought over from the New World. While 'smallpox' could leave a person horribly scarred and was fatal in about 30% of the cases, it usually passed in about a month or two. The great pox could last for over a year, and horrible, stinking, pus-filled lesions could cover the body for months, with no end in sight. While we have no good record of the mortality rate in early great pox victims, it actually appears that it was quite a bit less deadly than smallpox. Its reputation was still enough to scare Europe, even with the Bubonic plague still making the rounds. The old pox was renamed smallpox, and the great pox became the newest viral boogeyman.
[The Physicians] cared not even to behold it: so much less at the first to touch the Infected; for truly when it first begun, it was so horrible to behold. They had boils that stood out like Acorns, from whence issued such filthy stinking Matter, that whosoever came within the Scent, believed himself infected. The Colour of these was of a dark Green and the very Aspect as shocking as the pain itself, which yet was as if the Sick had laid upon a fire."
-- Ulrich von Hutten, 1519
The name syphilis originated from the poem Syphilis, sive Morbus Gallicus (Syphilis, or the French Disease), written by the doctor Girolamo Fracastoro in 1530. This poem described a shepherd named Syphilis, who was supposed to be the first sufferer of the disease. It wasn't until 1718 that syphilis started being used specifically as the name for the disease.
While the pox and syphilis are technically the same disease, they don't look very much alike. Modern syphilis causes small painless chancres and painless lesions that can sometimes even be mistaken for warts. While it is not a fun disease to have, and will sometimes break out into much more serious symptoms, it does not sound much like the great pox of 500 years ago.
This is an example of a disease adapting to its host; people covered with oozing gobs of pus don't tend to have as much sex as people with a light rash. Thus syphilis, being a sexually transmitted disease, had strong selective pressure to minimize its ugly symptoms. Eventually syphilis lost its pox and went undercover. And hence, the pox has disappeared, leaving us with simple syphilis.
The Western Medical Tradition: 800 B.C.-1800 A.D. by Lawrence I. Conrad, 1995
The great pox: the French disease in Renaissance Europe by Jon Arrizabalaga, John Henderson, and Roger Kenneth French, 1997