My guess as to why we have a better immune system would be our lifestyle: we are communal creatures, which probably causes viruses and bacteria to spread more quickly than in other, more sparsely populated creatures. Being in close contact with multiple other people would clearly increase chance of infection. This is well and good, but lots of other animals are also gregarious, so they may also have a better resilience to disease.
I would hazard a guess that we have better immune systems than these creatures as we are not nomadic - we stay in one place. As a result infection in the area may increase, especially before the times of sewage systems, due to viruses being able to stick around and reliably infect another individual.