These two characters serve as "character foils" in this particular story. Some of the similarities between Rainsford and Zaroff are that they are both "big game hunters" and "live for the hunt". In the past, they have both been exposed to hideous experiences in war. Both of them are characterized as being highly educated, shown through the fact that Zaroff is fluent in several languages, and Rainsford has written his own book.
The differences are more interesting. Zaroff, due to his experiences in war and work for the Czar, has little to no respect for human life. Rainsford finds this to be disgusting. Zaroff also has lost his passion for hunting animals, so, in accordance with his lack of respect to human life, he starts hunting humans. Rainsford still greatly enjoys hunting animals. Before being stranded with Zaroff, he was very excited about hunting in Rio. What we may see here is that Rainsford is an earlier version of Zaroff, not yet bitter and cold and heartless. However, by the end of the short story, he kills Zaroff after "winning the game". He justifies it by saying he is "still a beast at bay", but Zaroff had already awarded him victory. Rainsford killed him anyway.