It depends on whether you are asking about animals in general, or mammals specifically. Most non-mammalian animals have nuclei in their mature erythrocytes, or red blood, cells, but mammals, including humans, do not.
Comparing mammals, the main difference between the blood of two different species is the same as the difference between two people; the proteins that allow for self-identification. It is vital for any creature to be able to identify self versus non-self proteins so that the immune system can defend against non-self proteins without injuring proteins that are necessary for function. Some of the proteins are found embedded in the membranes of the blood cells; these include the ABO and Rh blood groups and the major histocompatibility complexes, and these vary from individual to individual as well as from species to species. Other proteins are dissolved in the blood plasma; these include the immunoglobulins.
So, basically, there is not a lot of difference among the blood of different species of mammals. In order to identify the origin of a sample of blood, one would need to do DNA testing.
I think the simplest thing which makes the difference between the human as well animals is antigens and antibodies. Antigens and antibodies are also used to distinguise between human blood groups.