That's an excellent question. Influenza and viruses like it are especially dangerous because they can infect animals as well. Strains of flu can inhabit birds, pigs, and humans (hence swine/bird flu). This means that a flu virus from both a human and a bird can infect a pig at the same time. When this happens, the two viruses can exchange genes in a process called antigenic drift. This can create a virus that has never been seen by anyone's immune system before, which can be devastating to a population. This is exactly what happened with the 1918-1919 pandemic, a new virus formed in a pig and then it was transferred to human populations. The ability to shuffle genes and create new viruses is what makes flu so difficult to deal with.
Viruses have high mutation rates because of a couple factors. RNA viruses generally mutate faster than DNA viruses since there is no proofreading system for RNA viral replication, which means more mistakes (mutations) make it to the next generation. Another reason viruses are able to mutate quickly is because they reproduce so fast. One virus infecting a cell can yield thousands of new virus with hundreds of new mutations.
Smallpox is a DNA virus, so it replicated relatively slowly and vaccines were able to prevent it in enough people to eradicate it before it could change. Measles and polio, however, are RNA viruses like HIV. So why were they able to be treated so efficiently with a vaccine? We were able to use the most successful strains of virus in a vaccine that was administered fast enough that the disease was unable to spread to more people and develop more mutations.
You might then ask why HIV does not have a vaccine. Well, HIV is different. Its mode of transmission, reservoirs in the body, ability spread very quickly across a population, and the fact that people can be exposed to it relatively frequently compared to other viruses, all contribute it infecting a population and mutating quickly. It is also difficult to create an effective vaccine for it without keeping the virus alive and using live HIV in a vaccine is not something many people are willing to test. HIV is just nasty. Every virus is different.
To sum up an answer to your question, flu viruses being able to infect different species and shuffle genes with all of those different strains sets it apart from other viruses.