There's a lot of confusion about different "poxes" in this post (which wasn't my intention, and now I feel bad), so here's a general overview (also, obligatory apology for messiness, this was written at like 1 AM):

Smallpox:

Smallpox, caused by variola virus, was a massive problem historically. It existed in the Western hemisphere for thousands of years (genetic evidence of smallpox has been found in Egyptian mummies from ≈1500 BCE, but it was probably around long before then), and it was introduced to the New World during the Columbian exchange, which had devastating consequences for indigenous populations (which were already suffering from colonialist violence, which made epidemics much worse than they already would've been). Historically, smallpox had a case fatality rate between 30-50%, and survivors were often left disfigured or permanently disabled (you've probably seen pictures of smallpox scars, but smallpox can also cause blindness and other complications). Importantly, smallpox only affects humans—it has no animal hosts—which is why it's one of the few infectious diseases to have been completely eradicated. As of May 8, 1980, it officially no longer exists outside of certain designated American and Russian laboratories. (There are, however, concerns that it could be used as a bioweapon, which is why the government still stockpiles smallpox vaccines and antivirals. I wrote my bioethics term paper on this exact issue, and incidentally, it's one of the major reasons why I believe that STEM majors should take ethics courses!)

There were two strains of variola virus: variola major and variola minor. Variola major was much more dangerous, with a much higher mortality rate; variola minor typically didn't cause severe disease. Fortunately, infection with one strain conferred immunity against the other. Both strains are now eradicated. (People sometimes confuse variola minor with other viruses like cowpox and horsepox, but they're different things.)

There were four clinical forms of smallpox: ordinary (classic smallpox, associated with the rash you usually see in pictures), modified (less severe, often occurred in vaccinated people who got infected anyway), malignant (caused a flat rash instead of the usual pustules, associated with immune dysfunction, almost always fatal), and hemorrhagic (caused severe bleeding, and also near-universally fatal.) All of the non-ordinary forms could be difficult to diagnose because they looked so different from typical smallpox. The less serious "modified" form was often confused with chickenpox, and the hemorrhagic form was sometimes assumed to be a completely different disease. Occasionally, historical sources will refer to hemorrhagic smallpox as "black pox," with or without an understanding that it's caused by the same virus as ordinary smallpox.

Other relevant viruses:

Cowpox, caused by cowpox virus (an orthopoxvirus similar to smallpox) causes mild disease in cows, humans, and several other animals. Infection with cowpox virus confers immunity to variola—Edward Jenner noticed this relationship and used material from cowpox lesions to inoculate people against smallpox.

Vaccinia virus, another orthopoxvirus, is the source of the modern smallpox vaccine. It's closely related to both cowpox and horsepox (weirdly, it's actually closer to horsepox), but it's distinct enough to be its own species. Infection usually causes mild symptoms, and, of course, confers immunity to smallpox.

Chickenpox is an entirely different thing. It's caused by the varicella-zoster virus, which is a herpesvirus, not a poxvirus at all! Infection with varicella-zoster does not confer immunity to smallpox or any other poxvirus—chickenpox is from a totally different family.

So why are the names so weird and confusing? Why is everything about all of this so weird and confusing?

There are multiple reasons for this, so bear with me.

Historically, a "pox" was any disease that caused a bumpy rash of pustles/blisters. Chickenpox, smallpox, and the other "poxes" all cause superficially similar rashes—thus the similar names. (Even though we know now that chickenpox comes from a completely different family, this wouldn't have been apparent before the dawn of modern medicine.)

Smallpox was given that name to differentiate it from syphilis, which was known as the "great pox" when it first appeared in Europe. (Fun[?] microbiology fact: There are debates about the origins of syphilis, but the most common theory holds that it originated in the New World, and Christopher Columbus brought it back to Spain. In that way, it's kind of the inverse of smallpox.) Historically, smallpox was also known by a variety of other names in different European, Asian, and African cultures. Again, this gets murky, because historical physicians sometimes struggled to distinguish between similar-looking-but-different diseases.

Other poxviruses are often named after the animals in which they were first identified. This is not a hard-and-fast rule, though, and it can sometimes be misleading (for example, monkeypox virus was first discovered in laboratory monkeys, but it more often affects rodents and other small mammals. The disease formerly known as "monkeypox" was recently renamed "mpox" because the name wasn't accurate.) Also, some poxviruses aren't named after animals at all! It's a weird and inconsistent system (but a lot of virus names are kinda weird and inconsistent).

Related to the above: We don't even know where the name "chickenpox" comes from. I mean, we know it was called a "pox" because it causes a pox-y rash, but we don't know where the "chicken" part originated. There are multiple theories about this, none of which are definitive. The disease itself has nothing to do with chickens.

Basically, a lot of the weirdness is a result of historical naming practices—people identified and named these diseases before modern virology existed, and those names stuck, so now we have similar names for superficially-similar-but-ultimately-different viruses, and names whose origins have been completely lost to time. Later, virologists muddied the waters further by naming newly-discovered poxviruses after the animals in which they were first seen, even when these animals aren't natural hosts or reservoirs of those viruses. It's a mess! And, again, all of this is complicated by the fact that some of these diseases were very hard to diagnose (or distinguish from one another) before modern medicine existed. Now, we can sequence viral DNA and figure out what's actually going on—which viruses caused which symptoms, whether those viruses were closely related, and whether being infected with one disease conferred immunity to another—but historical doctors and scientists didn't have those tools, so they were doing they best they could with very limited information, and that led to a lot of weirdness in terms of how these viruses were named and classified. Our current system inherited some of that weirdness, so here we are.

TL;DR: Poxvirus names are messy. Smallpox is caused by variola virus, which has two strains: variola major (the more severe one) and variola minor (less severe). Cowpox and vaccinia are different viruses in the same family, and being infected with one of them confers immunity to smallpox. Chickenpox isn't a poxvirus at all, but a herpesvirus—it just happens to cause a pockmark-y rash that looks superficially similar to smallpox pustules (and mild forms of smallpox were historically confused with chickenpox).

(P.S. none of this is super relevant to the average person, so don't feel bad if you didn't know any of it. Unless you are a history major inventing new conspiracies about smallpox, in which case you definitely should feel bad.)

Sources & further reading under the cut!

Waitress taking my order: Such arrogance! Not only do you presume to boast under the guise of being humble, but your order employs the most decadent of linguistic excesses - the tautology!

My date, who until recently thought "tautology" referred to the study of tensile strengths and upon learning her mistake compensated by reading through its Wikipedia article: That would be more correctly identified as a "pleonasm".

The editor I hired to curate my posts who styles himself as a sort of scheming court advisor: My liege, this one is getting away from us. The punchline loses much of its impact when the rest of the joke is derailed by this increasingly self-indulgent meta humour. Were it up to me, your Grace, which of course it is not, I would cut the others and leave myself as the only supporting character. You need noone else, Your Majesty...