The smallpox virus was (is, until the final stocks are destroyed) purely a human pathogen. That means that there were no animal reservoirs: if the disease is wiped out in humans, it's wiped out period - you don't need to go galloping through the outback trying to vaccinate wild foxes. Smallpox is also a rapidly progressive disease: that is, if you're infected today, you're going to be dead or recovered in a couple of weeks (I don't have the exact numbers handy, sorry). (Remember that vaccines are not, in general, effective after infection has been established: they are preventative rather than curative.) That means that if you go through a village and vaccinate everyone today, and you come back in three weeks and everyone is healthy, you can be confident that everybody in that village is immune (though see below). Contrast this to HIV, for example - if there ever is an effective preventive vaccine for HIV, and you vaccinate everybody in a village, you might not know for years if there are still persistently infected people. You can see how this has important implications for the spread of the disease.
The vaccine for smallpox is also unusual. To immunize against smallpox, people are infected with a live virus, vaccinia. Vaccinia is a poxvirus of unknown origin (it's most closely related to cowpox, but is distinct) which is related to smallpox. That means that if you're immune to vaccinia, you're probably immune to smallpox. There are a couple of reasons why a live vaccinia vaccine is so effective.
Firstly, live vaccines tend to be more effective than killed vaccines. A live virus is going to replicate, at least briefly: that means that the antigens are amplified. If you get 1000 vaccinia put into your arm scratch, then in three days - after several rounds of viral replication - there might be 1,000,000,000 viruses there.
Secondly, the live virus is more likely to be present in the same physical site as the actual pathogen. The immune system is, to some extent, physically partitioned - lymphocytes prefer to return to the same site over and over. If you stick a killed virus into your arms, it's mostly going to just sit there, and the general systemic immunity may be less effective. In the case of smallpox this might or might not have been critical; in some other cases, such as polio (which first infects the gut) getting local immunity can be very effective.
Thirdly, a live virus has far more antigens than a dead virus. If you put the viral particle in alone - as a killed vaccine does - then all the viral proteins involved in viral DNA replication, assembly, and so forth - those proteins that are expressed in infected cells, but not incorporated into the viral structure - are less likely to be in the equation. This can greatly increase the number of potential antigens to which the immune system can respond.
Fourthly, a live vaccine tends to stimulate a much broader range of immune responses. While killed antigens may trigger antibody responses and some forms of T-helper lymphocyte responses, live vaccine also may stimulate cytotoxic T lymphocytes, which can be very effective viral clearance mechanisms. I don't have space to explain exactly why; either take my word for it, or check out some immunology texts. Vaccinia also has some practical aspects that make it suited for use in the third world - it's very stable and can be quite readily grown, so you have less concern about, for example, refrigeration.
So you have several aspects of the vaccinia vaccine making it potentially potent. These overcame the two downsides of the vaccina vaccine. One is that vaccinia, while related, is not actually smallpox, so smallpox has many antigens which cross-react poorly with vaccina. But because vaccinia has so many different antigens, there's enough cross-reactivity to get effective immunity.
Another disadvantage of vaccina is that as vaccines go it's not really all that safe. For a less fearful threat than smallpox, vaccinia would have a hard time getting licensed. From reference [1]: "The use of vaccinia virus in recombinant systems may result in the occurrence of complications observed following smallpox vaccination. The results are presented of a large survey carried out in the USSR between 1968 and 1979. High complication rates are reported, particularly in older primary vaccinees; for example, there were 312.5 cases of neurological complications per million in those aged greater than or equal to 5 years." Mind you, there is presently work studying different vaccinia strains, and studying ways of making vaccinia less dangerous.
I don't have figures handy on the actual efficacy of the vaccina vaccine for smallpox. I suspect that the protection was probably in the same ballpark as the polio vaccines today. I'm sure it was not 100%; no vaccine is 100% effective. Of course, nothing is 100% effective.
So to summarize, the unique aspects of the smallpox virus (no animal carrier, no persistent disease) and the unique aspects of the vaccine for smallpox (a live virus, with all that implies) are probably the reasons why smallpox could be eradicated. The aspects of smallpox you've noted above are not the most important as far as the elimination went.
There are similar advantages with polio, and it's no coincidence that polio may also be eradicated by vaccination. As counterexamples, influenza has animal reservoirs, and it will never be eliminated by vaccination alone - barring a major advance in vaccine technology.
There are lots of other aspects I haven't touched on here; viruses have their own ways of resisting the immune system, and for some viruses these methods may affect vaccination efficacy. But that's much too complicated to cover here.
[Ref 1] Gurvich EB.
The age-dependent risk of postvaccination complications in vaccinees
with smallpox vaccine.
Vaccine. 10(2):96-7, 1992.