Animals and humans can get the same diseases but to my knowledge there are some diseases that we are immune to that they aren't and vice versa. Humans have had an advantage for the past century or so with having plumbing access to for the most part clean water and other means to prevent and fight diseases. Penicillin and amoxicillin are among pills that are made to fight and prevent diseases. Animals on the other hand have evolved in the wild where they have to rely on natural selection and how they have evolved to adapt in their environments to fight off sickness and disease.
Zoonotic diseases are diseases naturally able to transfer from animals to humans and vice versa. This shows that we share some of the same immune system capabilities as animals. For instance it is known that while sharks are able to get cancer their immune system prevents cancer/tumors from being able to grow.The recent Swine Flu outbreak is recent evidence of this resulting in many people becoming ill and around 12,000 actual deaths in the U.S. in the recent 2009-2010 outbreaks, and shows how diseases originate in animals and transfer to humans. AIDS however is a disease that cannot from my knowledge and research thus far, be contracted by animals except for in chimps. Is this because of our DNA similarities and because of the similar evolutionary path that we have gone down? Also, how has the evolutionary path of diseases influence how and who it effects between animals and humans? What about the sharks? Has evolving in the ocean over these millions of years gave them a sort of advantage from not letting cancer or tumors progress?
These following links show other diseases and how either vaccines are similar or animals and humans can contract similar diseases.