This is a good question, because it gets to one of the great paradoxes of the story. Both Zaroff and Rainsford act like animals in that they do not care for the huntee. In some ways, the difference between the two is not one of kind but of degree.
In the beginning of the story, when Rainsford's friend, Whitney posits that animals might have feelings, Rainsford dismisses this. Animal are animals, no feelings, no thoughts, in short, no rights. Rainsford simply says that there are two classes, the hunted and the huntee.
"Nonsense," laughed Rainsford. "This hot weather is making you soft, Whitney. Be a realist. The world is made up of two classes--the hunters and the huntees.
When Zaroff is introduced, Zaroff takes this point to its logical end as he includes humans into the mix. Some humans should be hunted.
If you look at these men from this perspective, they behave like animals. They do not care for others. For Rainsford animals have no rights; for Zaroff humans have no rights. In the end, Rainsford presumably kills Zaroff. This suggests that Rainsford is like Zaroff. Both, then, are bestial.